Algebra univers. 48 (2002) 479–484 0002-5240/02/040479 – 06 c Birkh¨ auser Verlag, Basel, 2002
Algebra Universalis
Flat and weakly flat projection algebras M. Mehdi Ebrahimi and M. Mahmoudi Abstract. The notion of a projection algebra was first introduced in [4] by Ehrig et al. as an algebraic version of ultrametric spaces. Computer scientists use this notion as a convenient means of algebraic specification of process algebras. Some algebraic notions regarding these algebras have been studied in [1], [2], [5]. The flat projection algebras have been investigated by the authors in [1]. Here we completely characterize flat and weakly flat (m-separated and separated) projection algebras.
1. Preliminaries One of the very useful categories in many branches of mathematics as well as in computer sciences is the category MSet of sets with an action of a monoid M on them. Here, we consider the category PRO of right N∞ -sets called projection algebras or projection spaces, where N∞ = N ∪ {∞} with m · n = min{m, n} and n < ∞, ∀n ∈ N. A separated projection algebra is one with an = bn (∀n ∈ N) ⇒ a = b An m-separated projection algebra for m ∈ N is one with an = bn (∀n ≤ m) ⇒ a = b Note that a projection algebra A is m-separated iff Am = A iff am = a for all a ∈ A. The categories of separated and m-separated projection algebras are denoted by PROs and PROm , respectively. Presented by M. Valeriote. Received November 19, 2001; accepted in final form December 1, 2002. 2000 Mathematics Subject Classification: 18B25, 18C05, 68B15. Key words and phrases: Projection algebra, separated projection algebra, flatness, weak flatness. The financial support from Shahid Beheshti University is gratefully acknowledged by the authors. 479
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2. Flatness In this section we study and characterize the flat, m-separated, and separated projection algebras. First recall from [1] or [2] that for projection algebras A, B, the tensor product A ⊗ B of A and B is defined to be the quotient set (A × B)/ ∼, where ∼ is the smallest congruence on A × B such that for each n ∈ N∞ , a ∈ A, b ∈ B, (an, b) ∼ (a, b)n and (a, bn) ∼ (a, b)n Definition 2.1. We call A ∈ PRO, a flat projection algebra if the functor A ⊗ − : PRO → Set preserves finite limits. The following two characterizations have been given in [1]. Theorem 2.2. For A ∈ PRO, the following are equivalent: (1) A is flat. (2) For every x, y ∈ A and s ∈ N∞ : (α) (x = y, xs = ys) ⇒ ∃z ∈ A, s , t ∈ N∞ (x = zs , y = zt ); (β) (xs = xt) ⇒ (xs = xt = x or s = t); (γ) ∃m ∈ N∞ (xm = ym). Theorem 2.3. For a projection algebra A, the following are equivalent: (1) A is flat. (2) (α ) For every x = y in A there exist z ∈ A, s , t ∈ N∞ (x = zs , y = zt ). (β) For every x ∈ A, s, t ∈ N∞ (xs = xt ⇒ xs = xt = x or s = t). Note that N∞ and any singleton {0} are flat as projection algebras. Theorem 2.4. A flat projection algebra is separated. Proof. Let A be a flat projection algebra. Let x = y in A with xn = yn for all n ∈ N. By (α ) there exist z ∈ A, s, t ∈ N∞ such that x = zs, y = zt. So, (zs)n = (zt)n for all n ∈ N. In particular, zs = (zt)s = (zs)t = zt; that is x = y which is a contradiction. Now we will characterize the flat projection algebras, flat m-separated and separated projection algebras. Definition 2.5. We call an element a of a projection algebra A fixed with respect to the action of n if n is the least natural number such that an = a, and a is called a fixed element if an = a for all n ∈ N. Notice that each nonempty projection algebra A has a fixed element. In fact for each a ∈ A, a1 is a fixed element.
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Lemma 2.6. A flat projection algebra has only one fixed element. Proof. By (α ) for every x = y in A there exist s, t ∈ N∞ and z ∈ A such that x = zs, y = zt, and so x(s · t) = y(s · t), hence x1 = y1. Therefore {x1 : x ∈ A} which is the set of all fixed elements of A has only one element. Lemma 2.7. Let A be a flat projection algebra. (a) If A is finite nonempty with |A| = m, then for each 1 ≤ n ≤ m, A has a unique fixed element with respect to the action of n. (b) If A is infinite then for each n ∈ N, A has a unique fixed element with respect to the action of n. Proof. Let a1 be the unique fixed element of A, which exists by the above lemma. Take x = y in A such that x, y = a1 . By (α ) there exist z ∈ A, s, t ∈ N∞ such that x = zs, y = zt. We have z = a1 , since x = y. Also, since x, y = a1 , s, t > 1. Therefore, s · t ≥ 2, so x(s · t) = y(s · t) implies x2 = y2. Thus there exists a2 ∈ A such that {x2 : x ∈ A − {a1 }} = {a2 }. Notice that if A is not a singleton set, a1 = a2 , by (β). Hence a2 is the unique fixed element of A with respect to the action of 2. Now, by this process, using induction we get the result. The following theorem determines the flat projection algebras. Theorem 2.8. A projection algebra A is flat iff A is isomorphic to an ideal of N∞ . Proof. First note that the set of all ideals of the monoid N∞ (the subsets which are closed under multiplication) is Ω = {↓k : k ∈ N∞ } ∪ {N∞ } where, for k ∈ N∞ , ↓k = {x ∈ N∞ : x < k}. Also notice that N = ↓∞ is an ideal of N∞ . Now two cases may occur: (1) A is finite nonempty with |A| = m. In this case, using part (a) of the above lemma, A = {a1 , a2 , . . . , am } where ak s = ak·s for 1 ≤ k ≤ m, s ∈ N∞ . Hence A ↓m. (2) A is infinite. In this case, using part (b) of the above lemma, there exists a subset B = {a1 , a2 , . . . , an , . . . } of A such that an s = an·s for n ∈ N, s ∈ N∞ . Now if B = A then A N. Otherwise there exists x ∈ A − B such that xn = an for all n ∈ N. Now, using 2.4, we conclude that such an x would be unique. So, A = B ∪ {x} and in this case A N∞ . Corollary 2.9. (a) A flat projection algebra is countable. (b) A flat projection algebra of cardinality m is m-separated. (c) If A is a flat m-separated projection algebra, then |A| ≤ m.
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Notice that the converse of part (c) (and trivially the converses of parts (a) and (b)) of the above corollary is not true. That is, an m-separated projection algebra A with |A| ≤ m is not necessarily flat. For example, the set {x, y, z} with actions given by xk = x, yk = zk = y for k ≤ 3 is a 4-separated projection algebra which is not flat. Remark 2.10. (a) There is no 1-separated flat projection algebra except the one point projection algebra. This is because the condition (α ) fails for these projection algebras, although condition (β) holds. (b) A 2-separated projection algebra A is flat if and only if for all x = y in A, x1 = y or y1 = x. Remark 2.11. (a) Although a subalgebra of a flat projection algebra is trivially flat, a quotient of a flat projection algebra is not necessarily flat. For example, N∞ /N 2 is not flat, while N∞ is flat. But one can show that if A/θ and the congruence θ are flat projection algebras then A is also flat. (b) In contrast to the case of modules, coproducts of flat projection algebras are not necessarily flat. For example, {0} is flat, while {0} {0} 2 = {0, 1} with identity actions is not flat. 3. Weak Flatness In this section using the results already obtained, we consider and completely determine weakly flat projection algebras (see also Stenstr¨om [6]). Also the relation between this notion and flatness is given. Definition 3.1. A projection algebra A is called weakly flat if the tensor product functor A ⊗ − : PRO → Set preserves equalizers and pullbacks. Similar to Theorem 2.6 we have: Theorem 3.2. A projection algebra A is weakly flat iff (α) (x = y, xs = ys) ⇒ ∃z ∈ A, s , t ∈ N∞ (x = zs , y = zt ), (β) (xs = xt) ⇒ (xs = xt = x or s = t). Remark 3.3. (a) Although flatness implies weak flatness there are weakly flat projection algebras, for example non singleton sets with identity actions, which are not flat. (b) The example given in part (a) also shows that the set of fixed elements of a weakly flat projection algebras is not necessarily singleton. Theorem 3.4. A weakly flat projection algebra which has only one fixed element is flat.
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Proof. It is enough to note that for x = y in a weakly flat projection algebra A, x1 = y1, by hypothesis, and so condition (α) implies (α ). Remark 3.5. For a projection algebra A define the equivalence relation ∼ by y ∼ z ⇔ y1 = z1 The equivalence class of a fixed element x with respect to ∼ is [x] = {z : x = z1}. In fact each class [x] of this relation contains a fixed element x1 and so is of the above form. Therefore A = {[x] : x is a fixed element of A}. Theorem 3.6. Each weakly flat projection algebra is a coproduct of flat projection algebras. Proof. The result follows from the fact that for each fixed element x, [x] is a subalgebra of A which has only one fixed element, and hence is flat by Theorem 3.4. Notice that the above theorem also shows that, in contrast to the case of flat projection algebras, the coproduct of weakly flat projection algebras is weakly flat. Theorem 3.7. Let A be a weakly flat projection algebra with cardinality m, x1 , . . . , xt be all the fixed elements of A, and |[xi ]| = ki , for i = 1, . . . , t. Then k1 + · · · + kt = m and each [xi ] is ki -separated and A is k-separated, where k = max{k1 , . . . , kt }. Proof. The first statement is because of the fact that these [xi ]’s form a partition of A and the second follows from Corollary 2.9(b). The last statement is clear because A is the union of these [xi ]’s. In contrast to the general case of M -sets (see [6]) we have: Theorem 3.8. A subalgebra of a weakly flat projection algebra is weakly flat. Proof. Let A be weakly flat and B ≤ A. Take x = y in B with xk = yk. By (α) there exist z ∈ A, s, t ∈ N∞ such that x = zs, y = zt. Put z = zn, where n = max{s, t}. Then z s = z(n · s) = zs = x and z t = y. Further z = zn is equal either to zs = x or to zt = y and so in both cases is in B. Notice that all the results given in this paper are true not only for N∞ -sets but also for M -sets where M is any commutative monoid all of whose elements are idempotent and which is totally ordered by the order ≤ given by x ≤ y ⇔ x · y = x. References [1] B. Banaschewski and E. Nelson, Tensor products and bimorphisms, Can. Math. Bull. Vol. 19 (1976), 385–402. [2] M. Mehdi Ebrahimi and M. Mahmoudi, Flat projection algebras, Proceedings of the Symposium of Categorical Topology (SoCat 94), Univ. Cape Town, 1999, pp. 113–120.
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[3] M. Mehdi Ebrahimi and M. Mahmoudi, Purity and equational compactness of projection algebras, Appl. Categ. Structures 9 (2001), 381–394. [4] H. Ehrig and H. Herrlich, The construct PRO of projection spaces: its internal structure, Lecture Notes in Computer Science 393 (1988), 286–293. [5] H. Ehrig, F. Parisi-Presicce, P. Boehm, C. Rieckhoff, C. Dimitrovici and M. Grosse-Rhode, Algebraic data type and process specifications based on projection spaces, Lecture Notes in Computer Science 332 (1988), 23–43. [6] E. Giuli, On m-separated projection spaces, Appl. Categ. Structures, 2 (1994), 91–99. [7] B. Stenstr¨ om, Flatness and localization over monoids, Math. Nachr. 48 (1970), 315–334. M. Mehdi Ebrahimi Department of Mathematics, Shahid Beheshti University, Tehran 19839, Iran e-mail :
[email protected] M. Mahmoudi Department of Mathematics, Shahid Beheshti University, Tehran 19839, Iran e-mail :
[email protected]
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