ALGEBRAIC PROPERTIES AND EXAMPLES OF INVERSE SEMIGROUPS

In this paper mainly we have obtained some important algebric properties of inverse semigroups.


Introduction :
In this paper we have obtained some important properties of inverse semigroups.In theorem 1.1 it is observed that any inverse semigroup is an equationally defineable algebra and any homomorphic image of an inverse semigroup is again an inverse semigroup, (which is actually the theorem 7.36 of Clifford and Preston, vol-II).Any subalgebra of an inverse semigroup is again an inverse semigroup and the direct product of any family of an inverse semigroups is an inverse semigroup.Further if S is an inverse semigroup, then a number of equivalent condition for x −1 y to be an idempotent of S(x, y ∈ S) are obtained(see theorem 1.3).It is also observed that if for any a, b ∈ S with a ∧ b exists and is equal to (a ∧ b) −1 .It is interesting to observe that if a −1 b is an idempotent of S, then for any c ∈ S, ac ∧ bc exists and is equal to (a ∧ b)c and ca ∧ cb also exists and is equal to c(a ∧ b) which are obtained in theorems1.4and 1.5.Further a number of equivalent conditions for any two elements a and a-1 of inverse semigroups to commute is obtained in theorem 1.6.Finally in theorem 1.7 it is obtained that, if S is an inverse semigroup in which a and a −1 commute ∀a ∈ S, then every congruence relation on E(S) can be extended to a congruence on S.
First we start with the following preliminaries: Def:0.1 : An element a of a semigroup S is said to be regular if there exists x ∈ S such that axa=a.A semigroup S is called regular if every element of S is regular.
Def:0.2 : Two elements a and b of a semigroup S are said to be inverse of each other iff aba = a and bab = b.
Def:0.3 : By an inverse semigroup we mean a semigroup in which every element has a unique inverse.
Def:0.4 : Let A be a universal algebra and B be a subalgebra of A then B is called a retract of A if there exists a homomorphism f : A → B such that f is restricted to B is identity on B. B is called retract and f is called a retraction mapping.
Def:0.5 : Let S be a semigroup and let a ∈ S, L(a) denotes the principal left Ideal S • a and R(a) denotes the principal right Ideal aS Proof : It is obvious.
Remark 1.1 : Since an inverse semigroup is equotionally definable from the general theory of W-algebras satisfying identity (a),(b),(c),(d).From the above we have the following corollaries Corollary 1.1 : An homomorphic image of an inverse semigroup is again an inverse semigroup (which is actually the theorem 7.36 of Clifford and Preston,vol-II).
Corollary 1.2 : Subalgebra of an inverse semigroup is again an inverse semigroup.
Corollary 1.3 : Direct product of any non-empty family of inverse semigroups is again an inverse semigroup.
The following examples establish the independency of the axioms mentioned in the above theorem.Then (S, •, −1) is a system satisfying (a),(c),(d) but not (b).Thus (b) is not a consequence of the remaining.
In the following theorem a number of equivalent conditions on the elements x and y of an inverse semigroup are obtained in order to x − 1y is an Idempotent of S.
Theorem 1.3 : If S is an inverse semigroup and x, y ∈ S, then the following are equivalent 1) x −1 y is an Idempotent of S 2) y −1 x is an Idempotent of S 3) yy − 1x ≤ y in the natural partial order on S 4) xx − 1y ≤ x in the natural partial order on S 5) G. L. B of x and y exists and is equal to xx −1 y 6) G. L. B of x and y exists and is equal to yy −1 x 7) G. L. B of x −1 and y − 1 exists and is equal to y − 1xx −1 8)G.L.Bof x −1 and y − 1 exists and is equal to The following example shows that there is an inverse semigroup in which f,g have g.l.b and f-1g is not an Idempotent.
In the following theorem a number of equivalent conditions on an inverse semigroup S are obtained in order to the elements a and a −1 commute ∀a ∈ S.
Theorem 1.6 : In an inverse semigroup S the following are equivalent; a and a −1 commute ∀a ∈ S, equivalently S in a Clifford semigroup ea = ae∀ idempotents e and ∀a ∈ S. la ra-1 is an endomorphism of S, where la is the inner left translation and ra −1 is the inner right translation.The mapping a → aa −1 is a retraction of S onto E(S).The mapping a → aa −1 is inverse preserving.L is a congruence relation.L = R. L ⊆ R. R ⊆ L L = R = D. S is a set union of groups.
Theorem 1.7 : If S is an inverse semigroup in which a and a −1 commute ∀a ∈ S, then every congruence relation on E(S) can be extended to a congruence on S.
Proof : Since E(S) is a retract of S and S is an universal algebra, the theorem follows.

Example 1 . 1 :
Let S = e, a, b Define '-1' in S by the compostion table as follows and define 0 − 1 0 by a − 1 = a, b − 1 = b and e − 1 = e.Then (S, •, −1) is a system satisfying (b),(c),(d) but not (a).Thus (a) is not a consequence of the remaining.Example 1.2 : Let S = e, a, b Define '-1' in S by the compostion table as follows and define 0 − 1 0 in S by a − 1 = b∀a ∈ S

Theorem 1 . 4 :
Let S be an inverse semigroup and let a −1 b be an idempotent element of S, for a, b ∈ S. Then for any c ∈ S, ac ∧ bc exists and is equal to (a ∧ b)c.Proof : Proof is clear Theorem 1.5 : Let a, b ∈ S and let a −1 b be an idempotent of S, then for any c ∈ S, ca ∧ cb exists and is equal to c(a ∧ b).