On the structure of hoops Introduction - CiteSeerX

On the structure of hoops W. J. Blok1 I. M. A. Ferreirim2 October 19, 1998 Abstract

A hoop is a naturally ordered pocrim (i.e., a partially ordered commutative residuated integral monoid). We list some basic properties of hoops, describe in detail the structure of subdirectly irreducible hoops, and establish that the class of hoops, which is a variety, is generated, as a quasivariety, by its nite members.

Introduction Residuated structures arise in many areas of mathematics, and are particularly common among algebras associated with logical systems. The essential ingredients are a partial order  , a binary operation of say multiplication
that respects the partial order, and a binary (left-) residuation operation ! characterized by c
a  b if and only if c  a ! b. In the logical context these represent a partial ordering of an algebra of truth values, (intensional) conjunction and implication, respectively. If the partial order is a semilattice order, and the multiplication the semilattice operation, we obtain the Brouwerian semilattices | the models of the conjunction-implication fragment of the intuitionistic propositional calculus. The well-known models of the conjunction-implication fragment of Lukasiewicz's many-valued logic are another example of a special class of residuated structures. The notion of residuated structure is in fact a very accomodating one, encompassing as well the models of weak logics such as the conjunction-implication fragment of Girard's linear logic | about the algebraic structure of which very little is known. Bosbach ([7, 8]) undertook the investigation of a class of residuated structures that were related to but considerably more general than the Brouwerian semilattices and the algebras associated with Lukasiewicz's calculus mentioned above. The requirement he added was that the partial order be natural ; in the commutative case (to which we will restrict ourselves in this paper) this means 1 Partly supported by a joint grant from JNICT, FACC, Proc.435.40/15160 and FLAD, Project 158/95. 2 Partly supported by research project Praxis XXI \A  lgebra e Matematicas Discretas".

1

that a  b if and only if there is a c such that a = b
c. Brouwerian semilattices as well as the models of many-valued logic satisfy this requirement, but the models of linear logic do not in general. He showed that the resulting class of structures can be viewed as an equational class, and that the class is congruence distributive and congruence permutable. In a manuscript by J. R. Buchi and T. M. Owens [9] devoted to a study of Bosbach's algebras, written in the mid-seventies, the commutative members of this equational class were given the name hoops . The manuscript is a rich source of ideas, but of a preliminary nature and was never published. One of its aims was to provide a precise description of the subdirectly irreducible hoops, and it did so under the added assumption that the multiplication be k-potent for some k. Although its treatment of the general case was awed, the characterization the authors aimed for is correct. The purpose of the present paper is to give an account of the characterization of the subdirectly irreducible hoops, based on Buchi's and Owens's ideas, with gaps lled in and corrections made as needed. Once the characterization is obtained we use it to show that the class of hoops can be seen as the natural common generalization of the classes of Brouwerian semilattices and of the models of the conjunction-implication fragment of Lukasiewicz's many-valued logic, which we call Wajsberg hoops . Not only does the variety of hoops contain both classes, we show that it can be obtained from the variety of Wajsberg hoops by a process analogous to the one by which the variety of Brouwerian semilattices can be obtained from Boolean algebras. More precisely, we associate with every variety of Wajsberg hoops an increasing sequence of varieties of hoops, each one obtained in a recursive way from the previous. If the class we start o with is the variety of conjunction-implication subreducts of Boolean algebras|the smallest non-trivial variety of Wajsberg hoops, termed Tarski algebras in [28] | the join of the associated sequence is the variety of Brouwerian semilattices. If the class we start o with is the variety of all Wajsberg hoops the join of the associated sequence is the variety of all hoops. In this sense the variety of hoops relates to the variety of Wajsberg hoops as does the variety of Brouwerian semilattices to the variety of Tarski algebras. As far as their conjunction-implication fragments are concerned, we may thus say that the logical system embodied in the class of hoops relates to Lukasiewicz's many-valued logic as does intuitionist logic to classical logic. As another application of the characterization of the subdirectly irreducible hoops we show that the variety, as well as several of its subvarieties, is generated, as a quasi-variety, by its nite members. It follows that its quasi-equational theory | and therefore the deductive system associated with it | is decidable. This is a slightly stronger version of a result stated in Buchi's and Owens's manuscript, but the strategy of our proof is due to Buchi and Owens. Finally, the characterization of subdirectly irreducible hoops provided the second author the necessary tool to show that the class of implicational subreducts of hoops is a variety, solving a problem raised in [40]; see [16] and [4]. Recently a syntactic proof was obtained by Kowalski [26]. 2

The notion of hoop has already found applications in a dierent context.

Hoops with normal multiplicative operators were introduced and studied in [6].

Many of the familiar varieties of logic, such as modal algebras, cylindric algebras, relation algebras, Heyting algebras and Wajsberg algebras can be viewed as varieties of hoops with normal multiplicative operators. Under certain natural restrictions on the operators [6] provides a characterization of the varieties of hoops with normal multiplicative operators that have equationally de nable principal congruences. Outline of paper. In Section 1 we recall some properties of residuated structures, outline the basic theory of hoops and introduce the important classes of k-potent hoops and Wajsberg hoops. Section 2 is devoted to the characterization of subdirectly irreducible hoops. The main result is Theorem 2.9, which says that a hoop is subdirectly irreducible if and only if it is an ordinal sum of any hoop and a subdirectly irreducible Wajsberg hoop. We de ne an operation on varieties of hoops reminiscent to that of Mal'cev's varietal power. It is used to generate from any variety of Wajsberg hoops an increasing chain of varieties of hoops. This, together with the characterization of subdirectly irreducible hoops, provides the tools to show that the variety of k-potent hoops is locally nite (Theorem 3.6) and that the variety of all hoops, although not locally nite, is generated, as a quasi-variety, by its nite algebras (Corollary 3.11).

1 Preliminaries

A structure A = hA;
; 1; i is a partially ordered monoid if hA;
; 1i is a monoid,  is a partial order on A, and for all x; y; z 2 A, if x  y, then x
z  y
z and z
x  z
y. A is integral if, for all x 2 A, x  1. A is residuated if for all x; y 2 A the set fz : z
x  yg contains a largest element, called the residual of x relative to y, and denoted by x ! y. A partially ordered, commutative, residuated and integral monoid hA;
; 1; i can be treated as an algebra hA;
; !; 1i, since the partial order can be retrieved via x  y i x ! y = 1. Such algebras will be referred to by the acronym pocrim. The class M of all pocrims satis es the following identities and quasi-identity:

(M1) x
1  x, (M2) x
y  y
x, (M3) x ! 1  1, (M4) 1 ! x  x, (M5) (x ! y) ! ((z ! x) ! (z ! y))  1, (M6) x ! (y ! z)  (x
y) ! z, 3

(M7) x ! y  1 & y ! x  1 ) x  y. Conversely, in every algebra hA;
; !; 1i satisfying (M1){(M7) we can de ne a partial order by setting x  y i x ! y = 1. This partial order makes hA;
; 1; i a commutative partially ordered monoid in which for all x; y 2 A x ! y is the

residual of x with respect to y. Hence, the class M of all pocrims is a quasivariety. Higgs [22] showed that the quasivariety M is not a variety. In addition to (M1){(M7), pocrims also satisfy the following properties: (M8) x ! x  1, (M9) x ! (y ! z)  y ! (x ! z), (M10) If x  y then y ! z  x ! z and z ! x  z ! y, (M11) x  (x ! y) ! y, (M12) x  y ! x. The quasivariety M of all pocrims is the equivalent algebraic semantics | in the sense of [5] | of the algebraizable deductive system SM below, as shown by J. Raftery and J. Van Alten [35]. The axioms of SM are: (B) ` ((p ! q) ! ((r ! p) ! (r ! q)), (C) ` (p ! (q ! r)) ! (q ! (p ! r)), (K) ` (p ! (q ! p)), (OK1) ` (p ! (q ! (p
q)), (OK2) ` ((p ! (q ! r)) ! (p
q) ! r) and inference rule Modus Ponens: (MP) p; p ! q ` q. As one can easily see, axioms (M5) and (M9) are the algebraic version of the axioms (B) and (C) of C.A. Meredith's BCK-logic, whereas (M12) is equivalent to x ! (y ! x)  1, which corresponds to the axiom (K), [33, 34]; the labels (B), (C) and (K) refer to the traditional names of corresponding combinators. The class BC K of BCK-algebras consists of all algebras A = hA; !; 1i satisfying (M3), (M4), (M5), (M7), (M8) and (M9). Since M satis es all of these, the class of f!; 1g-subreducts of algebras in M consists of BCK-algebras, that is, S(M f!;1g )  B C K . Conversely, every BCK-algebra is a subreduct of a pocrim and hence BC K = S(M f!;1g ); this was shown independently by Palasinski [32], Ono and Komori [31], and Fleischer [17]. Wronski [40] and Higgs [22] showed that B C K is not a variety. 4

Pocrims can be traced back to research undertaken in the rst half of this century on residuation in lattices of ideals of commutative rings with identity. In fact, if R is a commutative ring with identity 1 and I (R) is the monoid of ideals of R, with the usual ideal multiplication, ordered by inclusion, then for any two ideals I; J of R, the residual of I relative to J exists and is given by I ! J = fx 2 R : xI  J g. Hence, hI (R);
; !; Ri is a pocrim. J. R. Buchi and T. M. Owens [9] introduced a special class of pocrims which they called hoops. This class is the subject of the paper.

De nition 1.1 (i) A partially ordered commutative monoid A = hA;
; 1; i is naturally ordered if for all x; y 2 A, x  y i (9z 2 A) (x = z
y ): (ii) An algebra A = hA;
; !; 1i is called a hoop if it is a naturally ordered pocrim.

We denote the class of hoops by HO . Observe that in a hoop x  y if and only if x = (x ! y)
x. Moreover, in a hoop (x ! y)
x = (y ! x)
y holds for all x; y. As a consequence, every hoop is a meet-semilattice, with respect to its natural order, where the meet operation is given by x ^ y := (x ! y)
x, see [6]. The underlying ^-semilattice of any hoop is distributive, in the sense of G. Gratzer [20], i.e., whenever b ^ c  a, there exist b0  b, c0  c such that a = b0 ^ c0 | see [8, 9] . Conversely any pocrim satisfying the equation (x ! y)
x = (y ! x)
y is naturally ordered and hence a hoop. In fact, if x  y then x = 1
x = (x ! y)
x = (y ! x)
y; and if, for some z, x = z
y, it follows from (M6), (M8) and (M3) that x ! y = 1 and therefore x  y. Hence we may say that the class of hoops consists of those pocrims satisfying (M13) (x ! y)
x  (y ! x)
y. This remark is in fact a corollary of the following stronger result, due to B. Bosbach:

Theorem 1.2 [7] An algebra A = hA;
; !; 1i is a hoop if and only if hA;
; 1i is a commutative monoid that satis es the following identities: (M6) x ! (y ! z)  (x
y) ! z, (M8) x ! x  1, (M13) (x ! y)
x  (y ! x)
y.

5

Hence, the class of all hoops is a variety. Observe that the quasi-identity (M7) becomes super uous in the presence of (M13). Hoops, as well as pocrims, may be seen as algebras of logic. In fact,

Theorem 1.3 The variety of hoops is the equivalent algebraic semantics of the deductive system SHO , axiomatized by (B), (C), (K) and (OK1), (OK2), together with the axiom (H) ` ((p ! q)
p) ! ((q ! p)
q) and inference rule (MP).

Proof. The deductive system above is algebraizable since it is an axiomatic extension of the deductive system SM , and its algebraic semantics consists of all pocrims satisfying (x ! y)
x  (y ! x)
y, and hence, by symmetry, (M13) (x ! y)
x  (y ! x)
y | see [5, Theorem 2.17 and Corollary 4.9]. 2 A special class of rings provides one of the earliest examples of hoops: the pocrim of ideals of a Dedekind domain is naturally ordered, hence a hoop. Example 1.4 If R is a commutative integral domain with identity, R is a Dedekind domain if every nonzero ideal of R is uniquely representable as the product of nitely many prime ideals. Then the lattice-ordered monoid I (R) is naturally ordered, since, for any ideals I; J of R, if I  J then there are prime ideals P1; P2; : : : ; Pk such that I = P1
P2


Pk and J = Pi1
Pi2


Pit for some i1; i2 ; : : : ; it 2 f1; 2; : : : ; kg and therefore I = KJ, where K = Pj1
Pj2


Pjl and fj1 ; j2; : : : ; jl g = f1; 2; : : : ; kg n fi1 ; i2; : : : ; it g. Conversely, if I = KJ, for some K then I  J; therefore I  J if and only if I = KJ, for some K. A large class of Dedekind domains may be obtained p byptaking the integral closure of Z in any algebraic number eld, e.g., Z[ 10], Z[ 5i]. It is sometimes convenient to work with what we call dual hoops. A dual hoop is an algebra A = hA; +; ?
; 0i such that hA; +; 0; i is a partially ordered

commutative monoid, with identity 0, which is the least element of A, and for all x; y 2 A, x ?
y is the smallest element of the set fz : x  z+yg. Note that while in hoops the partial order satis es x  y i x = z
y, for some z 2 A (namely z = x ! y), the partial order in dual hoops satis es x  y i y = z+x, for some z 2 A. Thus if A = hA;
; !; 1i is a hoop then Ad = hA; +; ?
; 0i is a dual hoop, where x+y := x
y, x ?
y := y ! x and 0 := 1. Conversely, if A = hA; +; ?
; 0i is a dual hoop then Ad = hA;
; !; 1i is a hoop, where x
y := x+y, x ! y := y ?
x and 1 := 0. The classes of hoops and dual hoops are therefore term equivalent.

Example 1.5 Let G = hG; +; ?; 0; _; ^i be a lattice-ordered Abelian group, or Abelian `-group, for short (see [2] for detailed de nitions and properties). Let 6

P(G) be the positive cone of G, i.e, P(G) = fx 2 G : x  0g. On P(G) de ne the operation x ?
y = (x ? y) _ 0: Then P(G) = hP(G); +; ?
; 0i is a dual hoop. In particular, by taking the `group of the integers, Z = hZ; +; ?; 0; _; ^i and its positive cone, we obtain the dual hoop P(Z) = hP(Z); +; ?
; 0i. On the other hand, on the free monoid on one generator C1 = f1 = a0; a; a2; a3 ; : : : g, partially ordered by 1 = a0 > a = a1 > a2 > a3 >


we can de ne the operations an
am = an+m , and an ! am = amax(m?n;0) , for all n; m < !. Then the algebra C1 = hC1 ;
; !; 1i is a hoop, the dual of which is P(Z).

Example 1.6 Given an Abelian `-group G = hG; +; ?; 0; _; ^i and an arbitrary element u 2 G, u > 0, de ne on the set G[u] = fx 2 G : 0  x  ug the following operations: a +u b = (a + b) ^ u a ?
b = (a ? b) _ 0: Then hG[u]; +u; ?
; 0i is a dual hoop. This dual hoop may be seen as an instance

of a reduction of the action of the Gamma functor from Abelian `-groups with strong unit to MV-algebras, see [11, 29]. As in Example 1.5, consider the Abelian `-group of the integers, Z = hZ; +; ?; 0; _; ^i, an arbitrary positive integer m and the dual hoop Z[m] = hZ[m]; +m; ?
; 0i. Let Cm = f1 = a0 ; a; a1; a2; : : : ; amg with operations ak
am an = amin(k+n;m) ; ak ! an = amax(n?k;0), for all k; n  m. Then Cm = hCm;
am ; !; 1i is a hoop, the dual of which is Z[m]. Note that if a hoop is totally ordered then it satis es the identity (L) (x ! y) ! (y ! x)  (y ! x). We will now show that (L) holds for all hoops. This result will play a central role in the study of these algebras. In order to state and prove the result we need to introduce some notation. Let A be a hoop; for all x; y 2 A de ne x !0 y = y, and for any natural number +1 k y). k, x k! y = x ! (x !

Proposition 1.7 In any hoop the following holds:

(Lk) (x ! y) !k (y ! x)  y ! x, for every natural number k. Proof. To prove that (L) = (L ) holds we use an argument due to B. Bosbach [7]. Let c = (a ! b) ! a. We have a  c, by (M12), which in turn implies that c ! b  a ! b  c ! a by (M10) and (M11). Hence, (c ! b)
c  (c ! 1

7

a)
c  a, by de nition of residual. By (M13), (b ! c)
b = (c ! b)
c  a and so b ! c  b ! a. Thus, by (M9), (a ! b) ! (b ! a) = b ! ((a ! b) ! a) = b ! c  b ! a. The other inequality follows from (M12) and so we obtain (a ! b) ! (b ! a) = b ! a, as claimed. An easy argument by induction on k now completes the proof of (Lk ). 2

Filters and congruences. If A = hA;
; !; 1i is a hoop, we say that F  A is a lter of A if 1 2 F, F is closed under multiplication, and F is upward closed. It follows from (M12) that every lter of A is also a subuniverse of A. Moreover, one can easily check that, given X  A, the least lter of A containing X, i.e., the lter generated by X, denoted Fg(X), is

fb 2 A : a1 ! (a2 ! (: : :(at ! b) : : :)) = 1; for somea1; a2; : : : ; at 2 X; t < !g or, more simply Fg(X) = fb 2 A : a1
a2



at  b; for somea1; a2; : : : ; at 2 X; t < !g: n b = If, in particular, X = fag, then Fg(X) = Fg(a) = fb 2 A : a ! 1 for some n < !g = fb 2 A : an  b for some n < !g. It is easy to check that if  is a congruence of A then 1= is a lter of A. Moreover, the map  7! 1= establishes an order isomorphism between the lattice of congruences of A and its lattice of lters; the inverse of this map is F 7! F , where F = f(x; y) : (x ! y)
(y ! x) 2 F g is a congruence of A. If F is the lter associated with the congruence , we often write A=F for A=.

The order isomorphism between congruences and lters constitutes a useful tool in the study of hoops. B. Bosbach [8] made use of it in his proof that the variety HO is both congruence distributive (CD) and congruence permutable (CP), i.e., arithmetical. W. Cornish [13] showed that M(x; y; z) = ((y ! x) ! x) ^ ((z ! y) ! y) ^ ((x ! z) ! z) is a majority term and p(x; y; z) = ((x ! y) ! z) ^ ((z ! y) ! x) is a Mal'cev term for the variety of hoops, thus establishing congruence distributivity and congruence permutability as well. We use the order isomorphism between congruences and lters to show that HO has the congruence extension property (CEP).

Theorem 1.8 The variety of all hoops has the congruence extension property. Proof. Let A be a hoop and B a subhoop of A. It suces to show that for every lter F of B there exists a lter F 0 of A such that F 0 \ B = F. Let F be a lter of B and let F 0 be the lter of A generated by F; F 0 = Fg(F). Clearly, F  F 0 \ B. To see the converse, let b 2 F 0 \ B. Then there exist a ; a ; : : : ; an 2 F such that a
a


an  b. Since b 2 B and F is a lter of B hence upward closed and closed under multiplication, it follows that b 2 F. 2 1

2

1

2

8

k-potent hoops. In order to introduce the notion of k-potency, for any natural number k, we need some notation. Let A be a hoop; for all x 2 A de ne x0 = 1, and for any natural number k, xk+1 = xk
x. De nition 1.9 Let k < !; a hoop is called k-potent if it satis es the identity (k ) xk  xk+1. We denote the class of all k-potent hoops by HO (k). Observe that if a hoop is k-potent then it is l-potent for all l  k. Identity (k ) admits the following implicational version:

Proposition 1.10 In a hoop A, (i) x !k y = xk ! y for every x; y 2 A. (ii) A is k-potent if and only if A satis es the identity (k ) x !k y  x k! y. Proof. (i) Firstly note that x ! y = y = 1 ! y = x ! y, by (M4). Assume k y = xk ! y. Then, x k! y = x ! (x ! k by hypothesis of induction that x ! k k k y) = x ! (x ! y) = (x
x ) ! y = x ! y, by (M6). k y = xk ! y = xk ! y = x k! y, (ii) Assume A is k-potent. Then x ! by (i) and (k ). Thus A j= (k ). Conversely, assume A j= (k ). Then, for every x 2 A, xk ! xk = x !k xk = x k! xk = xk ! xk = 1, by (i) and (k ). Therefore, xk  xk . On the other hand, xk = xk
x  xk
1 = xk , hence xk  xk and therefore +1

0

0

+1

+1

+1

+1

+1

+1

+1

xk+1 = xk .

+1

+1

+1

+1

+1

+1

2 Example 1.11 Idempotent hoops, i.e, hoops which satisfy (1 ) : x  x2, are semilattices with respect to the operation of multiplication. They have been considered in the literature under the names implicative semilattices [30], and Brouwerian semilattices [25] and are the f^; !; 1g-subreducts of Heyting algebras. More generally, the nite chains Cm , introduced in Example 1.6, are mpotent hoops. On the other hand, C1 is not n-potent for any n.

Wajsberg hoops. The chains Cm ; m < ! and C1 satisfy the identity: (T) (x ! y) ! y  (y ! x) ! x.

Any hoop which satis es (T) is in fact a lattice, in which the join operation is given by x _ y := (x ! y) ! y. The identity (T) expresses the fact that _ is a commutative operation, and is therefore usually referred to as the commutative law, at least in the literature on BCK-algebras. 9

De nition 1.12 A hoop is a Wajsberg hoop if it satis es the commutative law (T). We denote the class of Wajsberg hoops by WHO .

If A is a Wajsberg hoop, and a 2 A, let A[a] = fx 2 A : x  ag. Since x ! y  y, A[a] is closed under !. De ne for x; y 2 A[a], x
a y = (x
y) _ a; and set A[a] = hA[a];
a; !; 1i. Then A[a] is a Wajsberg hoop with least element. We have Proposition 1.13 Let A be a Wajsberg hoop. Then A 2 ISPU (fA[a] : a 2 Ag).

Proof. For any a 2 A, let Ja be the set fx 2 A : x  ag (Ja is the principal ideal of A generated by a). Since for a; b 2 A Ja \ Jb = Ja^b , fJa : a 2 Ag has the nite intersectionQproperty. Let F be an ultra lter over A containing fJa : a 2 Ag and B = a2A A[a]=F . Then, the map A ?! B de ned by x 7! fx _ a : a 2 Ag=F is the desired monomorphism. 2 Corollary 1.14 C1 2 ISPU (fCm ; m < !g). Proof. It suces to observe that for m < !, Cm = C1 [m]. 2 By a bounded hoop we mean an algebra hA;
; !; 0; 1i such that hA;
; !; 1i is a hoop and 0  a, for all a 2 A.

Wajsberg hoops are closely related to the Wajsberg algebras of Font, Rodriguez and Torrens [18], which are algebraic models of Lukasiewicz's manyvalued logic. In fact, bounded Wajsberg hoops are termwise equivalent to Wajsberg algebras, see [6]. Moreover, it is easy to see that Wajsberg hoops which have a least element are exactly the f
; !; 1g-reducts of Wajsberg algebras. On the other hand, it follows from Proposition 1.13 that Wajsberg hoops are the f
; !; 1g-subreducts of Wajsberg algebras. Wajsberg algebras are de nitionally equivalent to C.C. Chang's MV-algebras, introduced in [10] and it is the formalism of MV-algebras that has been adopted by most authors. The implicational version of the de ning identity of Wajsberg hoops, (T), occurs in the list of axioms which Lukasiewicz conjectured to form a basis for the in nite-valued Lukasiewicz' propositional calculus. The rst known proof of Lukasiewicz' conjecture is attributed to Wajsberg, see [37, Ch. IV, 3]. However, his proof was never published. In the late 50's A. Rose and J.B. Rosser [36], by syntactical means, and C.C. Chang [11], by algebraic means, found a proof of Lukasiewicz' conjecture. 10

C.C. Chang's proof of Lukasiewicz' conjecture relied heavily on model-theoretic properties of Abelian `-groups. The latter are an essential source of examples of Wajsberg hoops. As we saw in example 1.5, given an Abelian `-group G = hG; +; ?; 0; _; ^i, its positive cone P(G) = fx 2 G : x _ 0 = xg is the universe of a dual hoop P(G) = hP(G); +; ?
; 0i, where x ?
y = (y ? x) _ 0. It is easy to see that both P(G)d and G[u]d, for any u 2 G; u > 0 satisfy the commutative law (T), hence they are Wajsberg hoops. Moreover, as a corollary to results obtained by C.C. Chang on MV-algebras, we can show

Theorem 1.15 Given a totally ordered Wajsberg hoop A with least element and jAj > 1, there exist a totally ordered Abelian `-group G and an element u 2 G, u > 0 such that A  = G[u]d.

Proof. Call 0 the least element of A and de ne for all x; y 2 A :x := x ! 0 and x + y := :x ! y. Then hA; +;
; :; 0; 1i is an MV-algebra and so is its dual A~ = hA;
; +; :; 1; 0i. Now use C.C. Chang's construction (see [11]) to obtain G and u from A~ . 2. By combining Proposition 1.13 and Theorem 1.15 we obtain Theorem 1.16 Every non-trivial totally ordered Wajsberg hoop can be embedded in a hoop of the form G[u]d, for some Abelian `-group G and some u 2 G, u > 0. Given an Abelian `-group G = hG; +; ?; 0; _; ^i and its positive cone P(G) = fx 2 G : x _ 0 = xg , the Wajsberg hoop P(G)d satis es also the cancellative

law :

(C) y ! (y
x)  x. Observe that any hoop satisfying (C) is cancellative as a monoid since if y
x = y
z, then x = z. Hoops satisfying (C) are called cancellative hoops and form a variety. We can appeal to a classical result (see for example [3, p. 321]) to see that if A is a cancellative hoop, then there is an Abelian `-group G such that A = P(G)d. In fact, we have Theorem 1.17 The functor G 7?! P(G)d , f 7?! f jP (G) establishes an equivalence between the category of Abelian `-groups and `-homomorphisms on the one hand and the category of cancellative hoops and homomorphisms on the other.

As seen in example 1.5, if Z is `-group of the integers hZ; +; ?; 0; _; ^i, then C1 = hf1 = a ; a; a ; : : : g;
; !; 1i, is isomorphic to P(Z)d . 0

2

Corollary 1.18 The variety of cancellative hoops equals ISPPU (C1 ). 11

Proof. It is well-known (see for example [23]) that the variety of Abelian `groups equals ISPPU (Z). The result follows from the fact the equivalence functor

of the previous theorem commutes with the operators I,S,P,PU . 2 Since the variety of cancellative hoops is generated by C1 and this is a Wajsberg hoop, cancellative hoops form a subvariety of the variety of Wajsberg hoops.

2 Subdirectly irreducible hoops The study of subdirectly irreducible hoops was rst undertaken by J. R. Buchi and T. M. Owens [9]. They completely characterized k-potent subdirectly irreducible hoops (see Corollary 2.10 below) and described, without providing satisfactory proof, the structure of subdirectly irreducible hoops in general. In this section, we give a correct proof of the description of subdirectly irreducible hoops, for which we use many ideas and auxiliary results of J. R. Buchi and T. M. Owens. Firstly, we describe simple hoops.

Lemma 2.1 Let A be a hoop. (i) A is simple if and only if A satis es (AL) For all a; b, if a =6 1 then there exists n < ! such that a !n b = 1. (ii) Every simple hoop satis es (M14) For all a; b b ! a = a implies a = 1 or b = 1. Proof. (i) Observe that A is simple if and only if for every a 2 A, if a =6 1 then Fg(a) = A. But, as we saw in the previous section, Fg(a) = fb 2 A : an  b; for some n < !g. Hence, (AL) follows. Conversely, if A satis es (AL) then, for all a 2 A; a = 6 1, every b 2 A belongs to Fg(a). Hence A is simple. (ii) Assume A is simple and let a; b 2 A be such that b ! a = a. Then for n0 a = 1, for some n a = a. Since A satis es (AL), if b = all n < !, b ! 6 1 then b ! n0 < !, and so a = 1. 2 Condition (AL) is known as the Archimedean Law, see [19]. Observe that any totally ordered Wajsberg hoop satis es (M14). Conversely, we have:

Proposition 2.2 Let A be a hoop satisfying (M14) For all a; b 2 A; b ! a = a implies a = 1 or b = 1. Then A is totally ordered and satis es (T), i.e., A is a totally ordered Wajsberg hoop.

12

Proof. The fact that A is totally ordered is a consequence of (L) (see [7]), as follows. For any a; b 2 A, (a ! b) ! (b ! a) = b ! a. Hence, by (M14), a ! b = 1 or b ! a = 1 and therefore, a  b or b  a. To see that A satis es (T), let a; b 2 A and assume without loss of generality that a < b; since (a ! b) ! b = 1 ! b = b, it suces to show that b = (b ! a) ! a. Observe that b ! a = ((b ! a) ! a) ! a by (M10) and (M11). Now [((b ! a) ! a) ! b] ! (b ! a) = = [((b ! a) ! a) ! b] ! [((b ! a) ! a) ! a] = [(((b ! a) ! a) ! b)
((b ! a) ! a)] ! a; by (M6) = [b ! ((b ! a) ! a)
b] ! a; by (M13) = [b ! ((b ! a) ! a)] ! (b ! a); by (M6) = 1 ! (b ! a); by (M11) = b ! a: Hence, by (M14) either b ! a = 1 or ((b ! a) ! a) ! b = 1. Since the assumption was that a < b, one has b ! a = 6 1 and therefore ((b ! a) ! a) ! b = 1, which implies (b ! a) ! a  b and hence, in view of (M11), (b ! a) ! a = b, as claimed. 2 Corollary 2.3 Every simple hoop is a totally ordered Wajsberg hoop. Proof. It follows immediately from Lemma 2.1(ii) and Proposition 2.2. 2 Example 2.4 The nite chains Cm ; m < !, as well as C1 are simple Wajsberg hoops.

We now turn our attention to subdirectly irreducible hoops. Recall that an algebra A is subdirectly irreducible if and only if A has a congruence  6=
that is contained in every non-zero congruence of A. We call  the monolith of A. Observe that if A is a subdirectly irreducible hoop, U = 1= is the unique minimal lter of A distinct from f1g. U is contained in every lter of A not equal to f1g. In the sequel, we will always denote this lter by U. Proposition 2.5 Let A be a subdirectly irreducible hoop with monolith . Then U is a subuniverse of A and hU;
; !; 1i is a simple totally ordered Wajsberg hoop.

Proof. We observed earlier that every lter is a subuniverse. Moreover, since U is the least lter of A distinct from f1g, U is generated, as a lter, by any n b= a 2 U n f1g, i.e., U = fb 2 A : an  b; for some n < !g = fb 2 A : a ! 1; for some n < !g. Hence, U satis es (AL). By Lemma 2.1 and Corollary 2.3, U is the universe of a simple, totally ordered Wajsberg hoop. 2 13

De nition 2.6 Let A be a subdirectly irreducible hoop with least lter U 6= f1g. An element a 2 A is said to be xed if for all u 2 U; u ! a = a. The set of xed elements of A is denoted by F. The set S = (A n F) [ f1g is called the support of U .

Proposition 2.7 Let A be a subdirectly irreducible hoop with monolith  , U =

1= and set of xed elements F . Then (i) (8a 2 A) [a 6= 1 ) (9u 2 U) u 6= 1 & a  u], (ii) U \ F = f1g, (iii) An element a 2 A is xed if and only if for some x 2 U n f1g x ! a = a.

m x = 1 for some m < Proof. (i) Let a 2 A; a 6= 1 and let Fg(a) = fx 2 A : a ! !g be the lter of A generated by a. Since Fg(a) 6= f1g, it follows that U  m x = 1. Choose Fg(a). Let x 2 U n f1g; then there exists m < ! such that a ! m x = 1 and let u = a m! ? x. Then m minimal with respect to the condition a ! m ? m x  u and u = 6 1 and so u 2 U n f1g. Moreover, a ! u = a ! (a ! x) = a ! x = 1 and therefore a  u. (ii) Let a 2 U \ F and u 2 U; u = 6 1. Since a is xed, u ! a = a. By Proposition 2.5, U is the universe of a simple hoop, and hence U satis es (M14) by Lemma 2.1(ii); since a; u 2 U, we have a = 1, as claimed. (iii) Assume a = 6 1 and x ! a = a for some x 2 U, x = 6 1. Let u 2 U be arbitrary; in order to show that a 2 F, we verify that u ! a = a. Since n a = a for every n < !. On the other hand, x ! a = a, we have x ! u  (u ! a) ! a and so (u ! a) ! a 2 U. Since U satis es (AL) and x = 6 1, n0 ((u ! a) ! a) = n0 ((u ! a) ! a) = 1 for some n < !. Hence, 1 = x ! x! n0 a) = (u ! a) ! a; thus u ! a = a. (u ! a) ! (x ! The converse is immediate, since U = 6 f1g. 1

1

0

2

Lemma 2.8 Let A be a subdirectly irreducible hoop, U its least lter distinct from f1g, F its set of xed elements and S the support of U . Let a 2 F; a 6= 1. Then (i) for all u 2 U a  u, (ii) for all u 2 U ua = a, (iii) for all b 2 A if b  a then b 2 F , (iv) for all b 2 A a ! b; b ! a 2 F , (v) for all b 2 S a  b; b ! a = a and ab = a.

14

Proof. (i) Since a 6= 1,m and U is the least lter distinct from f1g, U  Fg(a). Thus, given u 2 U, a ! u = 1 for some m < !. On the other hand, a ! u = (u ! a) ! (a ! u) by (L) = a ! (a ! u) since a is xed = a!u n u = a ! u, for every n < !. In particular, By induction, one derives a ! m 1 = a ! u = a ! u and so a  u. (ii) If u 2 U, then ua = u
(u ! a) = u ^ a = a, by (i). (iii) Let b  a. Let u 2 U be arbitrary. Then u ! b  u ! a = a and so (u ! b) ! a = 1. Now, (u ! b) ! b = 1 ! ((u ! b) ! b) = ((u ! b) ! a) ! ((u ! b) ! b) = (((u ! b) ! a)
(u ! b)) ! b by (M6) = ((a ! (u ! b))
a) ! b) by (M13) = ((au ! b)
a) ! b by (M6) = ((au ! b)
au) ! b since a is xed = 1 since (au ! b)
au  b: So u ! b  b  u ! b and therefore u ! b = b, yielding b 2 F. (iv) Let b 2 A . We want to show that both a ! b and b ! a are in F. Let u 2 U be arbitrary, then u ! (a ! b) = ua ! b; by (M6) = a ! b by (ii): Hence, a ! b 2 F. To show that b ! a 2 F, observe that u ! (b ! a) = b ! (u ! a) = b ! a and therefore b ! a 2 F. (v) Let b 2 S. If b = 1, all statements are trivially true. Hence we may assume without loss of generality that b = 6 1, i.e., b 2 A n F. By (iv), a ! b 2 F. Since b  a ! b and b 62 F, it follows from (iii) that a ! b = 1. Thus a  b. On the other hand, to show b ! a = a it suces to show that (b ! a) ! a = 1. Let u 2 U, u = 6 1. Then u ! a = a and u ! ((b ! a) ! a) = (b ! a) ! (u ! a) = (b ! a) ! a: So (b ! a) ! a is xed. By (iii), the fact that b  (b ! a) ! a and b 62 F, one gets (b ! a) ! a = 1 and so b ! a = a. Now, using b ! a = a and a  b we get ab = (b ! a)
b = a ^ b = a. 2 2

In order to present the structure of subdirectly irreducible hoops, we recall the construction of the ordinal sum of two hoops, A and B, denoted A  B. 15

If, for simplicity, we assume A \ B = f1g, de ne A  B to be the algebra with domain A [ B, and operations 1AB = 1A = 1B, 8 !A y for x; y 2 A; > < xx ! B y for x; y 2 B; x!y=> y for x 2 B; y 2 A; > :1 for x 2 A n f1g; y 2 B; and 8 A x; y 2 A; > < xx

B yy for for x; y 2 B; x
y = > y for x 2 B; y 2 A n f1g; :x for x 2 A n f1g; y 2 B:

If A \ B = 6 f1g replace A and B with isomorphic copies whose intersection is f1g and de ne their ordinal sum as above. This construction was rst studied by W. Cornish [13], who showed that A  B is a hoop and A and B are among its subalgebras. Also observe that B is a lter of the hoop A  B. One can easily check that if A and B are hoops then A  B is subdirectly irreducible if and only if B is subdirectly irreducible. We are now ready to describe subdirectly irreducible hoops.

Theorem 2.9 Let A be a subdirectly irreducible hoop, U its least lter distinct from f1g, F its set of xed elements and S the support of U . (i) F and S are subuniverses of A. Moreover, S is a lter of A. (ii) U  S, and S = hS;
; !; 1i is a subdirectly irreducible Wajsberg hoop. In particular, S is totally ordered. (iii) A = F  S. Proof. (i) 1 is xed, hence 1 2 F. To see that F is closed under multiplication and residuation, let a; b 2 F. If a = 1 then ab = 1b = b 2 F and a ! b = 1 ! b = b 2 F. If a 6= 1 then ab  b and so ab 2 F, by Lemma 2.8(iii); in addition, a ! b; b ! a 2 F, by Lemma 2.8(iv). This concludes the proof that F is a subuniverse of A. To show that S is also a subuniverse of A, let a; b 2 S. Since b  a ! b, then a ! b 2 S, by Lemma 2.8 (iii). To see that S is closed under multiplication, let a; b 2 S and a; b = 6 1 and assume, by way of contradiction, that ab 62 S. Then ab = 6 1, a 62 F, b 62 F and ab 2 F. Since a  b ! ab and a 62 F then b ! ab = 1 by Lemma 2.8(iii). However b ! ab = 1 implies b = ab 2 S, which contradicts our assumption that ab 62 S. Thus ab 2 S and S is a subuniverse of A as claimed. Since 1 2 S by de nition of S, in order to conclude that S is also a lter of A, it remains to observe that S is upward closed. This easily follows from lemma 2.8(iii).

16

(ii) U  S by Proposition 2.7 (ii) and the de nition of S. Since U is the least lter of A not equal to f1g it follows from CEP (see Theorem 1.8) that U is also the least lter of S not equal to f1g. Hence S is subdirectly irreducible. To prove that S is a totally ordered Wajsberg hoop, it suces to verify that it satis es (M14), by Proposition 2.2. Let a; b 2 S and assume that b ! a = a. If b 6= 1 then by Proposition 2.7 (i) there exists u 2 U such that b  u and u 6= 1. Then a  u ! a  b ! a = a , so a is xed, by Proposition 2.7(iii). Thus a 2 F \ S and hence a = 1, by Proposition 2.7(ii). (iii) This follows immediately from (i) together with Lemma 2.8 (v). 2 A typical subdirectly irreducible hoop may be depicted as in Figure 1: U



8 < F n f1g :

.1 9 > > = S > ;

 

Figure 1: A subdirectly irreducible hoop Observe that under the conditions of Theorem 2.9 one has F ' A=S. In the case of Brouwerian semilattices (i.e., idempotent hoops) the description given in Theorem 2.9 is well-known; the subdirectly irreducible Brouwerian semilattices are precisely the algebras of the form B  C1 , where B is any Brouwerian semilattice and C1 is the 2-element Brouwerian semilattice. More generally we have

Corollary 2.10 If A is a k-potent hoop, then A is subdirectly irreducible if and only if A = B  Cm for some m, 1  m  k, and some k-potent hoop B. Proof. One needs only to observe that in a k-potent subdirectly irreducible hoop A the support S of its minimal lter U coincides with U and is the universe of a k-potent subdirectly irreducible, totally ordered Wajsberg hoop S, by Theorem 2.9(ii). It follows that S is a simple k-potent Wajsberg hoop and therefore is isomorphic to some Cm , 1  m  k | see [12] for a detailed explanation. B is k-potent since it is a subalgebra of A. 2 For Wajsberg hoops the decomposition given in Theorem 2.9(iii) becomes trivial. Indeed, if A is any subdirectly irreducible Wajsberg hoop we have 17

U  S = A and F = f1g, and if in addition A is simple then U = S = A and F = f1g.

3 Generation by nite members In this section we use the characterization of the subdirectly irreducible hoops obtained in the previous section to acquire information about the variety of hoops and some important subvarieties. Given a class K of algebras, I K , HK , SK , PK , PU K denote respectively the classes of isomorphic images, homomorphic images, subalgebras, direct products and ultraproducts of members of K ; also, K SI and K F denote respectively the classes of subdirectly irreducible and nite members of K . For any algebra A, V(A) denotes the variety generated by A, i.e., HSP(A).

Theorem 3.1 The minimal varieties of hoops are precisely the minimal varieties of Wajsberg hoops, i.e., V(C1 ) and V(C1 ). Proof. Let V  HO , V a non-trivial variety. Then V contains a subdirectly irreducible algebra A and, by Theorem 2.9, A = F  S, with S a subdirectly irreducible Wajsberg hoop. Therefore V contains V(S), and hence a non-trivial variety of Wajsberg hoops. Thus V contains a minimal variety of Wajsberg hoops. It remains to show that the only two minimal varieties of Wajsberg hoops are V(C1 ) and V(C1 ). Let V be a non-trivial variety of Wajsberg hoops and A a subdirectly irreducible member of V. If  is the monolith of A and U = 1=, then U is the universe of a subalgebra of A, which is a simple totally ordered Wajsberg hoop, by Proposition 2.5. Hence V contains a simple totally ordered hoop U. Now, either U contains a least element or it doesn't. If U has a least element, say 0, then it is easy to check that f1; 0g is the universe of a subhoop of U isomorphic to C1. If U doesn't contain a least element, i.e., it is unbounded, let a 2 A, a 6= 1. We show that the one-generated monoid C = f1; a; a2; a3; : : : g is the universe of a subhoop of U, isomorphic to C1 . Since U is simple, for every element x 2 U; an  x for some natural number n. This fact, together with the unboundedness of U implies 1 > a > a2 > a3 > : : :. To see that C is closed under residuation, it suces to show that for any natural numbers n and k, if n < k then an ! ak = ak?n; indeed for n  k, an  ak and so an ! ak = 1. Given any natural numbers n; k, n < k, an
ak?n = ak and so ak?n  an ! ak , by de nition of residual. Since ak > ak+1, we have ak = an
ak?n 6 ak+1 and therefore, ak?n 6 an ! ak+1. The total order in A then implies that an ! ak+1 < ak?n. But U is a Wajsberg hoop, i.e., it satis es (T), and so for all x; y 2 U if x  y then y = (y ! x) ! x. In particular, an ! ak+1 < ak?n implies that 18

ak?n = ((ak?n ! (an ! ak+1)) ! (an ! ak+1 ) = ((ak?n
an) ! ak+1) ! (an ! ak+1 ) by (M6) = (ak ! ak+1) ! (an ! ak+1 ) = an ! ((ak ! ak+1) ! ak+1 ) by (M9) = an ! ak by (T) ; since ak+1  ak : This concludes the proof that C is a countable subuniverse of U, isomorphic to C1 . 2 The novelty of Theorem 3.1 resides in its proof, based on the description of subdirectly irreducible hoops. In fact, Amer [1] had already shown that V(C1 ) and V(C1 ) are the only two minimal varieties of hoops. Starting from any variety V WHO we de ne a sequence V1  V2  : : :  W n n + + V  : : :  V of varieties of hoops, where V = n
De nition 3.2 Let V be a variety of Wajsberg hoops. For any n < !, de ne the varietal power of V, Vn , as follows: (i) V0 is the trivial variety of hoops. (ii) Vn+1 =nHSP((Vn) ), where (Vn) is the class of algebras B  C such that B 2 V and C 2 V, C totally ordered. S Finally, V+ = ISPPU ( n 0. To show every n-generated algebra in W belongs to Vn, it suces to show every ngenerated subdirectly irreducible algebra in W belongs to Vn. So let A be such an algebra, and U, F and S as in Theorem 2.9. Then A = F  S and S 2 (W \ WHO)SI = VSI . Let X be a set of generators of A, jX j = n. Since f1g 6= U  S, and F is a subuniverse of A, we have X \ (S n f1g) 6= ;. Let f : A ?! F be the homomorphism which maps every element of S into 1 and every element of F into itself. Note that by the homomorphism theorem one obtains F ' A=S. Then F 2 W, and F is generated by f(X). Since 1 does not contribute to the generation, F is generated by ff(x) : x 2 X; f(x) 6= 1g as well, which is contained in ff(x) : x 2 X; x 62 S g. This last set has fewer than n elements and, by the induction hypothesis, it follows that F 2 Vn?1. Since S 2 VSI we conclude A 2 (Vn?1)  Vn . 2 19

Theorem 3.4 (i) HO = (WHO) . (ii) HO(k) = (WHO(k)) , for all k < !. +

+

Proof. Note that if W = HO or W = HO(k), for some k < !, and V = then V  W. Since any quasivariety is generated by its nitely generated members, the inclusion V  W follows immediately from Proposi-

W \ WHO ,

+

+

tion 3.3. 2 Part (ii) of Theorem 3.4 for k = 1 asserts that the variety of Brouwerian semilattices is the join of the iterated powers (WHO (1))n ; note that WHO (1) is HSP(C1 ), i.e., the variety of Boolean hoops. This representation was essentially obtained earlier by Kohler [24]. Theorem 3.4 suggests that the class of hoops relates to the class of Wajsberg hoops the same way the class of Brouwerian semilattices does to the class of Boolean hoops. Phrased in terms of deductive systems, the algebraizable deductive system SHO relates to Lukasiewicz's many-valued logic just like the intuitionistic propositional calculus relates to the classical propositional calculus | at least as far as their respective f
; !; 1g- fragments are concerned. We now want to show that the varieties HO (k), k < !, as well as HO , are generated by their nite algebras. We will use the following lemma.

Lemma 3.5 Let V  WHO be any variety and n < !. Then (Vn )SI  n +1

(V ) .

Proof. We argue by induction. The claim clearly holds for n = 0. Now assume it holds for k  n and let A 2 (Vn+1 )SI . Since HO is congruence distributive we mayQapply Jonsson's Lemma and so A 2 HS(D), where D 2 PU ((Vn) ); say D = ( i2I Bi  Ci)=F , where Bi 2 Vn, Ci 2 V, Ci totally ordered, and F a non-principal ultra lter on I.QBy properties of ultraproducts we see D  = B  C, Q  where B = i2I Bi=F , C  = i2I Ci =F ; indeed, B = fx 2 D : fi 2 I : x(i) 2 Bi g 2 Fg and C = fx 2 D : fi 2 I : x(i) 2 Ci g 2 Fg. Thus B 2 Vn, C 2 V and C is totally ordered. Now A 2 HS(D) and it is easy to see that therefore A = B0  C0 , where B0 2 HS(B),n C0 2 HS(C). Then B0 2 Vn, C0 2 V, C0 totally ordered and hence A 2 (V ) . 2 n +1 Observe that it follows from the lemma that in the de nition of V in De nition 3.2 the operator HSP can be replaced by ISP (or IPS ). Recall that a variety of algebras is said to be locally nite if its nitely generated members are nite. Theorem 3.6

HO (k) is locally nite, for all k < !. In particular HO (k) is generated by its nite members.

20

Proof. Since, by Proposition 3.3, every nitely generated algebra in HO(k)

belongs to (WHO (k))n , for some n < !, it suces to show (WHO (k))n is locally nite, for each n < !. For n = 0 this is trivial. Now assume (WHO (k))n is locally nite, for some n < !. If A 2 (WHO (k))n+1 is an m-generated subdirectly irreducible hoop, then in view of the previous lemma, A  = B  C, with B 2 (WHO (k))n and C 2 WHO (k) totally ordered. Moreover, since A is subdirectly irreducible, so is C and therefore C 2 (WHO (k))SI . Since B 2 H(A), B is m-generated as well, so jB j  jF(WHO (k))n (m)j = N. Furthermore C = Cl for some l  k, so jBj  N + k. We have thus a uniform upper bound on the size of the m-generated subdirectly irreducible hoops in (WHO (k))n+1 and it follows that (WHO (k))n+1 is locally nite. 2 Note that the upperbound on the size of the m-generated subdirectly irreducibles can be sharpened by using an argument similar to that used in the proof of Proposition 3.3. Clearly the variety HO of all hoops is not locally nite; indeed, the variety of Wajsberg hoops is not since C1 is 1-generated. HO is generated by its nite algebras, however, as we will see now. Let P = hP;
P; !P; 1Pi be a partial algebra of type (2; 2; 0); thus
P; !P are partial binary operations on P, and we will assume 1P 2 P. We call P a partial hoop if there is a hoop A such that P  A, 1P = 1A , and
P and !P are the restrictions of
A and !A respectively to P whenever those are de ned. We say then that P is a partial subhoop of A. We say that a class K of hoops has the nite embeddability property, (FEP), if for every nite partial subhoop P of some member A of K there is a nite member B of K such that P is a partial subhoop of B. If we denote the operation of forming partial subhoops by SP then a class K of hoops has the FEP if (SP (K ))F = SP (K F ). T. Evans proved that if a variety V has the FEP, then V = HSP(VF ) [15, Theorem 4]; his argument can be easily modi ed to show that, in fact, V = ISPPU (VF ) (see [16, Theorem 5.2]).

Lemma 3.7 Let V be a variety. If VSI has FEP, then so does V. Proof. Let A 2 V and let P be a nite partial subalgebra of A. Let f : A ?! Q i2I Ai be a subdirect representation of A, with Ai 2 VSI , i 2 I. Then (i  f)(P) is the universe of a partial subhoop of Ai , and by assumptionQalso of a nite subdirectly irreducible algebra in V, say A0i . Now f P : P ?! i2I A0i is an embedding and preserves all the existing operations. QSince P is nite, there is a nite subset I 0  I such that I  f P : P ?! i2I A0i gives the desired embedding into a nite algebra of V. 2 The following result is crucial. It is implicit in [38] and follows readily from Di Nola's description of MV-algebras in terms of ultrapowers of the MV-algebra 0

21

0

de ned on the unit interval [14]. We present a more algebraic proof, outlined earlier in [4]. Theorem 3.8 The class (WHO)SI has FEP. Proof. We use the fact that every totally ordered Wajsberg hoop can be embedded in a Wajsberg hoop of the form G[u]d, for some totally ordered Abelian `-group G, and some u 2 P(G) (see Theorem 1.16). Now with the nite partial subalgebra C of the Wajsberg hoop G[u]d we associate a universal Horn sentence C in the language of hoops such that for any totally ordered Abelian `-group H, the sentence C holds in Hd if and only if for no x 2 P(H), C is embeddable in H[x]d . Hence, the sentence C is not true in P(G)d . Since P(G)d is a cancellative hoop, it follows from Corollary 1.18 that the sentence C cannot hold in P(Z)d  = C1 either. Since the algebraic behavior of nitely many elements of P(Z)d is the same as their behavior considered as elements of Z[m]d , for m 2 Z large enough, we conclude that C can be embedded as a partial 2 algebra in Z[m]d  = Cm for some m 2 Z.

Theorem 3.9 The class (WHO)n, for each n < !, has FEP. Proof. By Lemma 3.7, it will suce to show (WHO)nSI has FEP. So let A 2 (WHO )nSI and let P be a nite partial subhoop of A. By Lemma 3.5, A  = B  C, where B 2 (WHO )n? and C 2 WHO totally ordered. More, since A is subdirectly irreducible, it follows also that C 2 (WHO )SI . Now let P be the partial subalgebra of P and of B with universe B \ P. By the induction hypothesis, P can be embedded in a nite algebra B0 2 (WHO )n? . Let P be the partial subalgebra of C and of P with domain C \ P. Since C is a subdirectly irreducible Wajsberg hoop it follows from Theorem 3.8 that P can be embedded in one of the hoops Cm , m < !. It is now not dicult to see that P itself can be embedded in B0  Cm 2 (WHO )n , via an extension of the embeddings of P into B0 and of P into Cm . 2 1

1

1

1

2

2

1

2

Corollary 3.10 The varieties (WHO)n; n < !, are generated, as quasivari-

eties, by their nite members; in fact, for n < !,

(WHO )n+1 = ISPPU fB  Cm : B finite; B 2 (WHO )n ; m < !g:

Corollary 3.11 The variety of hoops is generated, as a quasivariety, by its nite members.

The following lemma is the quasi-equational version of a well-known result on logical systems due to Harrop [21]: 22

Lemma 3.12 Let Q be a nitely axiomatizable quasivariety over a nite language. If Q is generated by a class of nite algebras, then the quasi-equational theory of Q is decidable. Proof. Since Q is nitely axiomatizable, the quasi-identities of Q can be recursively enumerated. Using the fact Q is nitely axiomatizable again, we see that the nite algebras of Q can be recursively enumerated. Since Q is generated by its nite algebras, every quasi-equation that fails to be a quasi-identity of Q fails in one of the nite algebras on the list. This allows us to recursively enumerate all quasi-equations that fail to be quasi-identities of Q. It follows that the set of quasi-identities of Q is recursive. 2 The next Corollary is an immediate consequence of the Lemma and Corollary 3.11: Corollary 3.13 The quasi-equational theory of the variety of hoops is decida-

ble.

Recall from section 1, that SHO is the deductive system which has the class of all hoops as its equivalent algebraic semantics. It follows from the last corollary that the set of rules of this deductive system SHO , i.e., the set of entailments of the form ? ` , with ? nite, is decidable.

References [1] K. Amer, Equationally complete classes of commutative monoids with monus, Algebra Universalis 18 (1984) 129{131. [2] M. Anderson and T. Feil, Lattice ordered groups, D. Reidel Publ. Co., Dordrecht, Holland, 1988. [3] G. Birkho, Lattice Theory, Amer. Math. Soc. Colloq. Publ. 25, 3rd ed., Amer. Math. Soc., Providence 1967. [4] W.J. Blok and I.M.A. Ferreirim, Hoops and their Implicational Reducts (Abstract), Algebraic Methods in Logic and Computer Science, Banach Center Publications 28 (1993), 219{230. [5] W.J. Blok and D. Pigozzi, Algebraizable Logics, Mem. Amer. Math. Soc. 396 (1989). [6] W.J. Blok and D. Pigozzi, On the structure of Varieties with Equationally De nable Principal Congruences III, Algebra Universalis 32 (1994) 545{ 608. [7] B. Bosbach, Komplementare Halbgruppen. Axiomatik und Arithmetik, Fund. Math. 64 (1969), 257{287. 23

[8] B. Bosbach, Komplementare Halbgruppen. Kongruenzen und Quotienten, Fund. Math. 69 (1970), 1{14. [9] J.R. Buchi and T.M. Owens, Complemented monoids and hoops, unpublished manuscript. [10] C.C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958),467{490. [11] C.C. Chang, A new proof of the completeness of the Lukasiewicz axioms, Trans. Amer. Math. Soc. 93 (1959), 74{80. [12] W.H. Cornish, Varieties generated by nite BCK-algebras, Bull. Austral. Math. Soc. 22 (1980), 411{430. [13] W.H. Cornish, BCK-algebras with a supremum, Math. Japon. 27 (1982), 63{73. [14] A. Di Nola, Representation and reticulation by quotients of MV-algebras, Recerche Mat. 40 (1991), 291{297. [15] T. Evans, Some connections between residual niteness, nite embeddability and the word problem, J. London Math. Soc. 1 (1969), 399{403. [16] I.M.A. Ferreirim, On Varieties and Quasivarieties of Hoops and their Reducts, Ph. D. Thesis, University of Illinois at Chicago, 1992. [17] I. Fleischer, Every BCK-algebra is a set of residuables in an integral pomonoid, J. Algebra 119 (1988), 360{365. [18] J.M. Font, A.J. Rodrguez and A. Torrens, Wajsberg Algebras, Stochastica 8 (1984), 5{31. [19] L . Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, Oxford, 1963. [20] G. Gratzer, General Lattice Theory, Birkhauser Verlag, Basel und Stuttgart, 1987. [21] R. Harrop, On the existence of nite models and decision procedures for propositional calculi, Proc. Cambridge Philos. Soc. 54 (1958), 1{13. [22] D. Higgs, Dually residuated commutative monoids with identity element do not form an equational class, Math. Japon. 29 (1984), 69{75. [23] W. C. Holland, A. H. Mekler and N. R. Reilly, Varieties of lattice-ordered groups in which prime powers commute, Algebra Universalis 23 (1986), 196{214. 24

[24] P. Kohler, Varieties of Brouwerian algebras, Mitt. mathem. Seminar Giessen, 116, 1975. [25] P. Kohler, Brouwerian semilattices, Trans. Amer. Math. Soc. 268 (1981), 103{126. [26] T. Kowalski, A syntactic proof of a conjecture of Andrzej Wronski, Rep. Math. Logic 28 (1994), 81{86. [27] A.I. Mal'cev Multiplication of classes of algebraic systems, Sib. Math. J., 8 (1967), 254{267. [28] A. Monteiro, Cours sur les algebres de Hilbert et de Tarski, Instituto Mat. Univ. Del Sur, Bahia Blanca, 1960. [29] D. Mundici, Interpretation of AF C -algebras in Lukasiewicz sentential calculus, J. Funct. Anal. 65 (1986) 15{63. [30] W. Nemitz, Implicative semi-lattices, Trans. Amer. Math. Soc. 117 (1965), 128{142. [31] H. Ono and Y. Komori, Logics without the contraction rule, J. Symbolic Logic 50 (1985), 196{201. [32] M. Palasinski , An embedding theorem for BCK-algebras, Math. Seminar Notes Kobe Univ. 10 (1982), 749{751. [33] M. Palasinski and A. Wronski, Eight simple questions concerning BCKalgebras, Reports on Mathematical Logic 20 (1986), 87{91. [34] A.N. Prior, Formal Logic, Clarendon Press, Oxford, second edition, 1962 [35] J.G. Raftery and J. Van Alten, On the algebra of noncommutative residuatrion: polrims and left residuation algebras, Math. Japonica 46 (1997), 29{46. [36] A. Rose and J.B. Rosser, Fragments of many-valued statement calculi, Trans. Amer. Math. Soc. 87 (1958), 1{53. [37] A. Tarski (translated by J.H. Woodger), Logic Semantics Metamathematics, Hackett Publ. Co., Indianapolis, Indiana, second edition, 1983 [38] R. Wojcicki, On matrix representations of consequence operations of Lukasiewicz's sentential calculi, Zeitschr. f. math. Logik und Grundlag. d. Math. 19 (1973), 239{247. [39] A. Wronski, BCK-algebras do not form a variety, Math. Japon. 28 (1983) , 211{213. 25

[40] A. Wronski, An algebraic motivation for BCK-algebras, Math. Japon. 30 (1985) , 187{193. Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago Chicago, IL 60607 U.S.A.

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Departamento de Matematica Faculdade de Ci^encias Universidade de Lisboa 1700 Lisbon Portugal