Natural Partial Order on Some Class of Semigroups - m-hikari

International Journal of Algebra, Vol. 8, 2014, no. 10, 479 - 484 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2014.4554

Natural Partial Order on Some Class of Semigroups P. G. Romeo and M. S. Jisha Dept. of Mathematics Cochin University of Science and Technology Kochi, Kerala, India c 2014 P. G. Romeo and M. S. Jisha. This is an open access article disCopyright tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract The natural partial order on a regular semigroups was introduced by K. S. S. Nambooripad in 1980. Later natural partial order on an arbitrary semigroup was also introduced by H. Mitsch. In this paper we describe the natural partial order on the class of semigroups Mn (R) and the semigroup L(V ) of linear operators on a finite dimensional vector space V . Mathematics Subject Classification: 20M10 Keywords: Semigroup, Natural partial order A set S is a groupoid with respect to a binary operation if for every pair of elements a, b ∈ S there is an element a · b ∈ S which is the product of a by b. A groupoid S is a semigroup if the binary operation on S is associative. An element a ∈ S is called regular if there exists an element a ∈ S such that aa a = a, if every element of S is regular then S is a regular semigroup. An element e ∈ S such that e · e = e is called an idempotent and the set of all idempotents in S will be denoted by E(S). The class of regular semigroups in which the idempotents commute is the class of inverse semigroups. A subset I of a semigroup S is called left (right, two sided) ideal if SI ⊆ I (IS ⊆ I, SIS ⊆ I). In the study of the structure of semigroups ideals and idempotents play vital role. A binary relation on a set X is a partial order if it is reflexive, antisymmetric and transitive. For any semigroup S the set of idempotents E(S) is a partially

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ordered set with respect to the partial order defined by e≤f

⇐⇒

ef = f e = e ∀ e, f ∈ E(S).

It is possible to extend this ordering from E(S) to the whole of S, defining it by means of the multiplication of S and such partial order is termed as the natural partial order on S. 1. Preliminaries In the following we assume familiarity with the basic terminology and result of semigroup theory as in [3] and [1], however we recall a few definitions and results needed in the sequel. Let S be a semigroup and a ∈ S then the smallest left ideal containing a is Sa∪{a} and is called the principal left ideal generated by a, written as S 1 a. Also aS 1 is the principal right ideal generated by a and S 1 aS 1 the principal two sided ideal generated by a. Next we define certain equivalence relations on semigroups using these principal ideals. Definition 1.1. Let a, b be elements in a semigroup S, then • aLb if and only if they generate the same principal left ideal. ie., S 1 a = S 1 b. • aRb if and only if they generate the same principal right ideal. ie., aS 1 = bS 1 . • aJ b if and only if they generate the same principal two sided ideal. ie., S 1 aS 1 = S 1 bS 1 . The intersection of R and L is of great importance and is denoted by H and their join by D. These equivalence relations are termed as Green’s equivalences and are significant in the study of semigroups. Note that in a regular semigroup each L - class and each R - class contain idempotents, ie., regular semigroups may be described as those class of semigroups in which each L - class and each R - class contain idempotents and inverse semigroups are those semigroups where each L - class and each R - class contain a unique idempotent. Proposition 1.1. In a semigroup S the relations Lx ≤ Ly ⇐⇒ S 1 x ⊆ S 1 y and Rx ≤ Ry ⇐⇒ xS 1 ⊆ yS 1 are partial orders on S/L and S/R respectively and Hx ≤ Hy ⇐⇒ Lx ≤ Ly and Rx ≤ Ry is a partial order on S/H. An important class of semigroups for which a natural partial order was found is the class of inverse semigroups (cf. V.Vagner 1952). Lemma 1.1. Let S be an inverse semigroup. For x, y ∈ S define x≤y

⇐⇒

x = ey

for some e ∈ E(S).

Then ≤ is a partial order on S whose restriction to E(S) coincides with the partial order on E(S).

on some class of semigroups

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Proposition 1.2. Let S be a regular semigroup. For x, y ∈ S define x≤y

⇐⇒

Rx ≤ Ry

and x = ey

for some e ∈ E(Rx ).

Then the relation ≤ is a partial order on S whose restriction to E(S) coincides with the partial order on E(S). 2. Natural partial order on Mn (R) The set Mn (R) of n × n matrices with entries in R with respect to matrix multiplication is a regular semigroup. In the following we always denote the semigroup Mn (R) by S and for any A in S, col(A) [row(A)] denotes the column space [row space] of A. Lemma 2.1. Let A, B ∈ S. Then RA ≤ RB if and only if col(A) ⊆ col(B), which further implies A R B if and only if col(A) = col(B). Proof. Suppose RA ≤ RB in S, then we have AS 1 ⊆ BS 1 and hence there exists some X ∈ S such that BX = A. Now a system Bx = a where ⎛ ⎞ ⎞ ⎛ consider ⎞ ⎛ a1 x1 b11 b12 . . . b1n ⎜ a2 ⎟ ⎜ x2 ⎟ ⎜ ⎟ ⎟ ⎜ ⎜ b21 b22 . . . b2n ⎟ ⎟ , x=⎜ . ⎟ , a=⎜ . ⎟ B=⎜ ⎜ ⎟ ⎟ ⎜ ⎝ . . . . . . ⎠ ⎝ . ⎠ ⎝ . ⎠ bn1 bn2 . . . bnn xn an ie., b11 x1 + b12 x2 + .... + b1n xn = a1 b21 x1 + b22 x2 + .... + b2n xn = a2 ...................... bn1 x1 + bn2 x2 + .... + bnn xn = an which implies ⎛ b11 ⎜ . x1 ⎜ ⎝ . bn1

⎞

⎛

⎞ ⎛ b12 b1n ⎟ ⎜ . ⎟ ⎜ . ⎟ + x2 ⎜ ⎟ ⎜ ⎠ ⎝ . ⎠ + . . . + xn ⎝ . bn2 bnn

⎞

⎛

⎞ a1 ⎟ ⎜ ⎟ ⎟ = ⎜ . ⎟ ⎠ ⎝ . ⎠ an

That is the column matrix (ai ) is a linear combination of columns of B and hence BX = A implies every columns of A is a linear combination of columns of B which implies col(A) ⊆ col(B). Conversely col(A) ⊆ col(B) implies every columns of A is a linear combination of columns of B, thus BX = A for some X ∈ S, which implies AS 1 ⊆ BS 1 and hence RA ≤ RB . Now it is easy to see that A R B if and only if col(A) = col(B). Proposition 2.1. Let A, B ∈ S. Then A ≤ B if and only if col(A) ⊆ col(B) and A = EB for some idempotent matrix E in RA .

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Proof. From the definition of natural partial order on regular semigroups, we have A ≤ B if and only if RA ≤ RB and A = EB for some idempotent matrix E in RA . By Lemma 2.1 RA ≤ RB if and only if col(A) ⊆ col(B). Hence A ≤ B if and only if col(A) ⊆ col(B) and A = EB for some idempotent matrix E in RA . Now it is enough to show the existence of idempotents in every R-class of S. We have A R B if and only if col(A) = col(B), thus the R-class of a matrix A in S is regarded as a subspace of Rn , because every column space of a n × n matrix is a subspace of Rn . Let A ∈ S be of rank n, then A ∈ GLn (R) and since GLn (R) is a group it has a single R-class. Thus all rank n matrices form a single R-class and it contains an idempotent matrix In . The only possible matrix with column space “zero vector” is the zero matrix, that is the zero matrix forms a single R-class (say R0 ) and is an idempotent matrix. Now consider RA the R-class of the matirix A with rank(A) = r < n, then there exists a subspace V of Rn with dim(V ) = r < n and col(A) = V . Since subspaces fixe R-classes, we denote each R-class by RV for each V ⊆ Rn . Now it is enough to show that there exists matrices E in RV with E 2 = E. Note that for V ⊂ Rn there exists U ⊂ Rn such that Rn = V + U where U is the complementary subspace of V . Hence x ∈ Rn can be expressed as x = xv + xu where xv ∈ V and xu ∈ U. Define the projection map, E : Rn −→ V

by E(x) = xv .

Here E is a projection matrix onto V implies ran(E) = V and hence E ∈ RV . Now E 2 (x) = E(E(x)) = E(xv ) = xv = E(x) for all x ∈ Rn implies E 2 = E. Therefore E is an idempotent matrix. Hence every R-class contains idempotents. 3. The natural partial order on L(V ) Heinz Mitsch in [4], generalized the definition of natural partial order on regular semigroups to arbitrary semigroups in 1994. The following propositions are from [4]. Proposition 3.1. For any semigroup (S, .) and a, b ∈ S a relation ≤ defined by : a ≤ b ⇐⇒ a = xb = by, xa = a for some x, y ∈ S 1 is a partial order and is called the natural partial order (n.p.o.) on S. The restriction of ≤ to the subset E(S) of all idempotents of S (if it exists) coincides with the usual ordering on E(S). Proposition 3.2. For an arbitrary semigroup S and its n.p.o. the following are equivalent : 1. a ≤ b 2. a = xb = by, ay = a for some x, y ∈ S 1 3. a = xb = by, xa = a = ay for some x, y ∈ S 1

on some class of semigroups

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Now we proceed to describe the natural partial order on the semigroup S = L(V ) of all linear operators on a finite dimensional vector space V . Proposition 3.3. Let A, B ∈ S. A ≤ B if and only if A = C ◦ B = B ◦ D and C|R(A) = I|R(A) for some C, D ∈ S 1 where R(A) = range(A). Proof. From Proposition 3.1 we have A ≤ B ⇔ A = C ◦ B = B ◦ D and A = C ◦ A for some C, D ∈ S 1 . Note that A = C ◦ A ⇔ A(x) = C(A(x)) ∀ x ∈ V ⇔ C|R(A) = I|R(A) . Therefore, A ≤ B if and only if A = C ◦ B = B ◦ D and C|R(A) = I|R(A) for some C, D ∈ S 1 . Lemma 3.1. Let A, B ∈ S. If A ≤ B then ran(A) ⊆ ran(B) and ker(B) ⊆ ker(A). Proof. Suppose A ≤ B. Then A = C ◦ B = B ◦ D and C|R(A) = I|R(A) for some C, D ∈ S 1 . Now A = B ◦ D =⇒ ran(A) = {A(x)/x ∈ X} = {B(D(x))/x ∈ X} ⊆ {B(x)/x ∈ X} = ran(B). Assume that x ∈ ker(B) ⇒ B(x) = 0 which implies C(B(x)) = 0 and hence A(x) = 0. Thus x ∈ ker(A). Therefore ker(B) ⊆ ker(A). Proposition 3.4. Let A, B ∈ S. If A ≤ B and A is invertible then A = B. Proof. Assume that A ≤ B and A is invertible. Hence A = C ◦ B = B ◦ D and C|R(A) = I|R(A) for some C, D ∈ S 1 . Since A−1 exists A is one-one and onto ie., ran(A) = V and hence C = I on V . Therefore A = C ◦ B = I ◦ B = B. Lemma 3.2. Let A, B ∈ S. If A ≤ B and B is invertible then there exists idempotents C and D in S 1 such that A = C ◦ B = B ◦ D and C|R(A) = I|R(A) . Proof. Suppose A ≤ B. Then A = C ◦ B = B ◦ D and A = C ◦ A for some C, D ∈ S 1 . Now it is easy to see that C ◦B = A = C ◦A = C ◦(C ◦B) = C 2 ◦B that is C ◦ B = C 2 ◦ B. Existence of B −1 implies C ◦ B ◦ B −1 = C 2 ◦ B ◦ B −1 , thus C = C 2 . Similarly B ◦ D = B ◦ D 2 and hence D = D 2 . Therefore C and D are idempotents. Proposition 3.5. Let I be the identity operator and let 0 be the zero operator. Then we have the following. 1. 0 ≤ A for every A ∈ S. 2. A ≤ I if and only if A is an idempotent. Proof. 1. For every A ∈ S, 0 = 0 ◦ A = A ◦ 0, 0 = 0 ◦ 0. i.e. choose C = D = 0 implies 0 ≤ A for every A ∈ S.

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2. Let A ≤ I. Since I is an idempotent A is also an idempotent. Conversely assume that A is an idempotent, that is A = A ◦ A. Choose C = D = A. Thus A = C ◦ I = I ◦ D, A = C ◦ A which implies A ≤ I. Therefore A ≤ I if and only if A is an idempotent. References [1] A. H. Cliﬀord and G. B. Preston. The Algebraic Theory of Semigroups, Volume 1, Math. Surveys of the American. Math. Soc.7, Providence, R. I, 1961. [2] P.A.Grillet. Semigroups, An Introduction to the Structure Theory, Marcel Dekker, Inc, 1995. [3] J. M. Howie. An Introduction to Semigroup Theory. Academic Press, London, 1976. [4] H. Mitsch. Semigroups and their natural order. Math. Slovaca. 44, No. 4: 1994, 445-462. [5] K.S.S. Nambooripad. Structure of regular semigroups-I: Fundamental regular semigroups, Semigroup Forum 9: 1975, 354-363. [6] K.S.S. Nambooripad. The natural partial order on a regular semigroup. Proc. Edinburg Math. Soc., 23: 1980, 249-260.

Received: May 11, 2014