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Toplogical properties of inverse semigroups.

Introduction

In this paper mainly we have obtained certain intresting topological properties induced by the natural partial order of an inverse semigroup is obtained.

It is observed that the subalgebra E(S) is an algebraic retract of S, where S is an inverse semigroup iff a and a-1 commute for all aeS. But it is interesting to observe that E(S) is always a topological retract of S which is obtained in theorem 2.2. Since the direct product of any family of inverse semigroups is also an inverse semigroups. It is natural to examine whether the topology on the product semigroup is the product topology.

In corollary 2.3.1 it will be shown that a topology on the product semigroup is finner than the product topology and also observed that the two topologies need not be the same because of example 2.1 further it is observed that the map (a,b)[right arrow][ab.sup.-1] from SxS into S where S is an inverse semigroup, is continous with respect to the semigroup topology.

First we start with the following preliminaries

Def 1.1: By an inverse semigroup we mean a semigroup in which every element has unique inverse. Equivalently a regular semigroup in which idempotents commute.

Def 1.2: Let A be a universal algebra and B be a sub algebra of A. Then B is called retract of A iff there exists a homomorphism f: A[right arrow]B such that 'f' restricted to B is identity on B.

Def 1.3: A binary relation [rho] defined on S where S is an inverse semigroup is called a partial equivalence relation on S iff [rho] is symmetric and transitive.

Theorem: 1.4[2] Let H be an inverse sub semigroup of the inverse semigroup S. Then [H.sub.[omega]] is a closed sub semigroup of S.

Now we start with the following theorem

Theroem 2.1: Let {[S.sub.[alpha]]|[alpha] [member of] I} be a family of inverse semigroups then the natural partial order on the product semigroup [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the direct product ordering on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof: If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then f [less than or equal to] g iff [fg.sup.-1] = [ff.sup.-1] iff [alpha]([fg.sup.-1]) = [alpha]([ff.sup.-1]) for every [alpha] [member of] I iff (([alpha])f)[(([alpha])g).sup.-1] = (([alpha])f)[(([alpha])f).sup.-1] for every [alpha] [member of] I iff ([alpha]) f [less than or equal to] ([alpha]) g for every [alpha] [member of] I Thus f [less than or equal to] g in the natural partial order on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in the direct product order on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theroem 2.2: If 'S' is an inverse semigroup, then f: S [right arrow] S by a[right arrow][aa.sup.-1] is a continous function and infact it is a topological retraction of S onto E(S).

Proof: Let 'A' be a closed subset of 'S'.

Now [f.sup.-1](A) = {a [member of] S/f(a) [member of] A} = {a [member of] S/[aa.sup.-1] [member of] A}

Let x [[member of].sup.-] [f.sup.-1] [(A).sup.-] so that x [greater than or equal to] a for some a [member of] [f.sup.- 1](A) and hence x [greater than or equal to] a for some a such that f(a) [member of] A. Hence x [greater than or equal to] a for some a such that [aa.sup.-1] [member of] A and hence [xx.sup.-1] [greater than or equal to] [ax.sup.-1] = [aa.sup.-1] since x [greater than or equal to] a for some [aa.sup.-1][member of]A therefore [xx.sup.-1] [member of][sup.-]A[sup.-] = A as A is closed and [aa.sup.-1][member of]A so that [f.sup.-1](A) is contained in [sup.-][f.sup.-1](A)[sup.-] as A is contained in [sup.-]A[sup.-] and hence

[sup.-][f.sup.-1](A)[sup.-] = [f.sup.-1](A).

Therefore [f.sup.-1](A) is a closed subset of S and that f: S [right arrow] S is continous. Further if e[member of]E(S) then f(e)= [ee.sup.-1] = ee = e that f is a retraction of S onto E(S).

Theroem 2.3: Let {[S.sub.[alpha]]|[alpha] [member of] I} be a family of inverse semigroups and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the [alpha]th projection map of [PI] [S.sub.[alpha]] on to [S.sub.[alpha]]. Then [P.sub.[alpha]] is continous with respect to the semigroup topologies.

Proof: Let F be a closed subset of [S.sub.[alpha]] and let x [member of] [sup.- ][P.sub.[alpha].sup.-1](F)[sup.-] so that x [greater than or equal to] a for some a [member of] [P.sub.[alpha].sup.-1](F) and hence x [greater than or equal to] a for some [P.sub.[alpha]](a) [member of] F therefore ([alpha])x [greater than or equal to] ([alpha])a for some ([alpha])a [member of] F and hence ([alpha])x [member of] F so that [P.sub.[alpha]](x) [member of] F and hence x [member of] [P.sub.[alpha].sup.- 1](F) thus [sup.-][P.sub.[alpha].sup.-1](F)[sup.-] is contained in [P.sub.[alpha].sup.-1](F), but [P.sub.[alpha].sup.-1](F) is contained in [sup.- ][P.sub.[alpha].sup.-1](F)[sup.-] as A is contained in [sup.-]A[sup.-] and [P.sub.[alpha].sup.-1](F) = [sup.-][P.sub.[alpha].sup.-1](F)[sup.-]. Therefore [P.sub.[alpha].sup.-1](F) is a closed subset of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. and thus [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is continuous.

Corollary 2.3.1: The semigroup topology on the product semigroup [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is finer than the product topology on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Example: 2.1 Let S be the two element chain {0,1} and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (Sn = S for each n).

Define (n)f = 0, if n is odd = 1, if n is even

Let G be the open set {g[member of]G/g [less than or equal to] f}. Suppouse U is a basic open set in the product topology containing f and contained in G. Since U is a basic open set, there exist [m.sub.1][m.sub.2],,...[m.sub.k] such that [P.sub.n](U) = [S.sub.n] = S for all n [not equal to] [m.sub.1][m.sub.2],,... [m.sub.k]. Let n be an odd integer such that n [not equal to] [m.sub.1][m.sub.2],,...[m.sub.k] then [P.sub.n](U) = S and the map 'g' is defined by (m)g = 0 for all m[not equal to]n [m.sub.1][m.sub.2],,...[m.sub.k] and [m.sub.i](g) = [m.sub.i](f), 1 [less than or equal to] i [less than or equal to] k and (n)g=1 [member of] U. Then 1 = (n)g [less than or equal to] (n)f = 0 which is a contradiction.

Theorem 2.4: Suppouse S is an inverse semigroup. Let f: S x S [right arrow] S be defined by f(a,b) = a.b then f is continuous with respect to the semigroup topologies.

Proof: Let F be a closed subset of S and let (x,y) [member of] [sup.-][f.sup.- 1](F)[sup.-] so that (x,y) [greater than or equal to] (a,b) for some (a,b) [member of] [f.sup.-1](F) and hence x [greater than or equal to] a, y [greater than or equal to] b for some f(a,b) [member of] F. Therefore xy [greater than or equal to] ay [greater than or equal to] ab, since a [less than or equal to] x,b [less than or equal to] y for some ab [member of] F and hence xy [member of] F so that f(x,y) [member of] F and hence (x,y) [member of] [f.sup.-1](F).

Thus [sup.-][f.sup.-1](F)[sup.-] is contained in [f.sup.-1](F) so that [sup.- ][f.sup.-1](F)[sup.-] = [f.sup.-1](F).

Theorem 2.5: If S is an inverse semigroup then f: S [right arrow] S is defined by f(a) = [a.sup.-1] is a homeomorphism.

Proof: It is obvious.

Theorem 2.6: Let S,T be inverse semigroups and f,g are continuous isotone maps from S into T. Define h: S [right arrow] T by h(a) = f(a).g(a) then h is continuous with respect to the semigroup topologies.

Proof: Let F be a closed subset of T and let x [member of] [sup.-][h.sup.- 1](F)[sup.-] so that x [greater than or equal to] a for some a [member of] [h.sup.-1](F) and hence x [greater than or equal to] a for some h(a) [member of] F. Therefore f(x) [greater than or equal to] f(a) and g(x) [greater than or equal to] g(a) for some h(a) [member of] F and hence f(x).g(x) [greater than or equal to] f(a).g(a) for some f(a).g(a) [member of] F, so that h(x) [member of] F and hence x [member of] [h.sup.-1](F).

Thus [sup.-][h.sup.-1](F)[sup.-] is contained in [h.sup.-1](F) and hence [sup.- ][h.sup.-1](F)[sup.-] = [h.sup.-1](F). so that [h.sup.-1](F) is a closed subset of S. Thus h: S [right arrow] T is continuous.

Theorem 2.7: If S is an inverse semigroup then they map (a,b) [right arrow] [a.b.sup.-1] is continuous with respect to semigroup topology.

Theorem 2.8: Let S be an inverse semigroup and let a [member of] S, define f: S [right arrow] SxS by (b)f=(a,b) then f is continuous with respect to semigroup topologies.

Theorem 2.9: Let S be an inverse semigroup and let a [member of] S, then the map b [right arrow] a.b of S into S is continuous with respect to the semigroup topologies.

Proof: It is clear by using theorems 2.4 and 2.8.

Theorem 2.10: If S and T are inverse semigroups and f: S [right arrow] T is homomorphism and T is an inverse sub semigroup of T such that [f.sup.-1](K) is nonempty then [f.sup.-1](K) is an inverse sub-semigroup of S.

Proof: It is obvious.

Corollary 2.10.1: Let S and T be inverse semigroups and f: S [right arrow] T is a homomorphism.

If e is an idempotent of T and [f.sup.-1](e) is nonempty then [f.sup.-1](e) is an inverse sub semigroup of S.

Theorem 2.11: Let a be an element of an inverse semigroup S and let H be a closed subset of S such that F = {x [member of] S/ax [member of] H} then F is closed.

Proof: Clearly f: x[right arrow]ax is continuous. Now [f.sup.-1](H) = {x [member of] S/f(x) [member of] H} = {x [member of] S/ax [member of] H} = F therefore F is closed.

Theorem 2.12: Let H be a closed inverse sub semigroup of S, and let e [member of] H be an idempotent element of S and also [aa.sup.-1] = [a.sup.-1]a for all a [member of]S, then F = {x [member of] S/ex [member of] H} is closed inverse sub semigroup of S.

Proof: First it is observed that f: x[right arrow]ex is a homomorphism.

Further, [f.sup.-1](H) = {x [member of] S/ex [member of] H} is nonempty. By using theorems 2.10 and 2.11 F is closed inverse sub semigroup of S.

Theorem 2.13: If [H.sub.1][H.sub.2] are closed inverse sub semigroups of S, then [H.sub.1] is contained in [H.sub.2] iff [PI] [H.sub.1] is contained in [PI][H.sub.2].

Proof: Suppose [H.sub.1] is contained in [H.sub.2], let (x,y) [member of] [PI] [H.sub.1] so that [xy.sup.-1] [member of] [sup.-][H.sub.1][sup.-] = [H.sub.1] as [H.sub.1] is closed and hence [xy.sup.-1] [member of] [H.sub.2] so that(x,y) [member of] [PI] [H.sub.2].

Thus [PI] [H.sub.1] is contained in [PI][H.sub.2].

Conversely suppouse [PI] [H.sub.1] is contained in [PI][H.sub.2] and let x [member of] [H.sub.1] so that [x.sup.-1]x and [xx.sup.-1]x [member of] [H.sub.1].

Therefore, x = C imply that x[([x.sup.-1]x).sup.-1] [member of] [H.sub.1]. Hence (x, [x.sup.-1]x) [member of] [PI][H.sub.1] so that (x, [x.sup.-1]x) [member of] [PI] [H.sub.2] and hence x[([x.sup.-1]x).sup.-1] [member of] [H.sub.2] imply that x([x.sup.-1]x) [member of] [H.sub.2].

Thus x [member of] [H.sub.2].

References

[1] A.H. Clifford and G.B. Preston: The algebraic theory of semigroups, American Mathematical soc.providence, Vol.1 1964.

[2] A.H. Clifford and G.B. Preston: The algebraic theory of semigroups, American Mathematical soc.providence, Vol.1 1967.

K.V.R. Srinivas (1) and R. Nandakumar (2)

(1) Professor and (2) Asst. Professor, (1,2) Mathematics

(1,2) Regency Institute of Technology, Yanam-533464

(1) E-mail: srinivas_kandarpa06@yahoo.co.in

(2) E-mail: rnandakumar2001@yahoo.com
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Author:Srinivas, K.V.R.; Nandakumar, R.
Publication:International Journal of Difference Equations
Date:Jun 1, 2011
Words:2333
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