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Regular Semigroups with Strongly E-Inverse Transversals

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Regular Semigroups with Strongly E-Inverse Transversals International Mathematical Forum, Vol. 7, 2012, no. 15, 737 - 743 Regular Semigroups with Strongly E-inverse Transversals Yuxiao Gao and Yingchun Wang Jiyuan Vocational and Technical College Luoyang, Henan, 454650, P.R. China [email protected] Abstract In this paper, we introduce a new inverse transversal E-inverse transver- sal. We discuss some natures of the regular semigroup with E-inverse transversals and strongly E-inverse transversals. Furthermore this pa- per gives the smallest inverse semigroup congruence and the greatest idempotent separating congruence on regular semigroups with strongly E-inverse transversals. Keywords: regular semigroups; strongly E-inverse transversals; the small- est inverse semigroup congruence 1 Introduction and preliminaries In 1982, Blyth and McFadden introduced regular semigroups with inverse transversals in [6], this type of semigroup has attracted much attention. An 0 inverse subsemigroup S of a regular semigroup S is an inverse transversal if 0 |V (x) S | = 1 for any x ∈ S, where V (x) denotes the set of inverses of x. In this case, the unique element of V (x) S0 is denoted by x0 and (x0)0 is denoted by x00. Let S be a regular semigroup with set E(S) of idempotents, and let e, f ∈ E(S). Then the set S(e, f), defined by S(e, f)={g ∈ E(S) | ge = fg = g, egf = ef} is called the sandwich set[3] of e and f. Let S be a regular semigroup with an inverse transversal S0. Then S0 is called a multiplicative inverse transversal[6] of S if x0xyy0 ∈ E(S0) for all x, y in S; S0 is called a weakly multiplicative inverse transversal[4] of S if (x0xyy0)0 ∈ E(S0) for all x, y in S; S0 is called an E-inverse transversal[5] of S if x0xyy0 ∈ E(S) for all x, y ∈ S. We will use the following Lemmas: 738 Yuxiao Gao and Yingchun Wang Lemma 1.1 [2] A regular semigroup S with an inverse transversal S0 is ortho- dox if and only if (xy)0 = y0x0 for all x, y in S. Lemma 1.2 [3] Let S be an orthodox semigroup, then the set E(S)inS forms a band under multiplication, this is expressible as a semilattice Y of E E rectangular band Eα (α ∈ Y ), and V (e)=Je for any e ∈ E(S), where Je is the rectangular band Eα containing e. Lemma 1.3 [1] Let S be an orthodox semigroup with an inverse transversal S0, then for any e ∈ E(S) the equivalence relation μ = {(x, y) ∈ S × S | x0ex = y0ey, xex0 = yey0} is the greatest idempotent separating congruence on S. Lemma 1.4 [3]A regular semigroup S is orthodox if and only if for any x, y ∈ S V (x) ∩ V (y) = ∅ =⇒ V (x)=V (y). Lemma 1.5 [3] Let S be an orthodox semigroup, then the equivalence relation γ = {(x, y) ∈ S × S | V (x)=V (y)} is the smallest inverse semigroup congruence on S. In this paper, we introduce a new inverse transversal E-inverse transversal. We discuss some natures of the regular semigroup with E-inverse transversals. 2 The main results An inverse transversal S0 of a regular semigroup S is called a strongly E-inverse transversal if x0xyy0 ∈ E(S), xx0y0y ∈ E(S) for all x, y in S. Before investigating the natures of this semigroup ,we pause to consider a example: suppose semigroup S = {e, f, g}, we obtain the Cayley table efg e ee e f eff g eg g Then the semigroup S is orthodox with strongly E-inverse transversals S0 = {e, f} and S0 = {e, g}. Regular semigroups with strongly E-inverse transversals 739 Infact the operation is associative: (ef)g = e(fg)=e;(eg)f = e(gf)=e;(fe)g = f(eg)=e, (fg)e = f(ge)=e;(ge)f = g(ef)=e;(gf)e = g(fe)=e. Since fgf = ff = f, gfg = gg = g, f ∈ V (g), but fef = e, efe = e, e∈ / V (f); geg = e, ege = e, e∈ / V (g). Thus S0 = {e, f} and S0 = {e, g} are inverse subsemigroups of S and every element has the unique inverse in S0 = {e, f} and S0 = {e, g} for all x in S, then it is easily to verify that S is orthodox with strongly E-inverse transversals S0 = {e, f} and S0 = {e, g}. Lemma 2.1 Let S be a regular semigroup with an E-inverse transversal S0. Then (1) e0 ∈ E(S0) for any e ∈ E(S); (2) S0 is a weakly multiplicative inverse transversal. Proof. (1) Let e ∈ E(S), then e0 = e0ee0 = e0eee0 ∈ E(S), and e0 ∈ S0,thus e0 ∈ E(S0). (2) Since S be a regular semigroup with an E-inverse transversal S0,thus x0xyy0 ∈ E(S) for any x, y ∈ S. Then by(1), we get (x0xyy0)0 ∈ E(S0), and so S0 is weakly. Lemma 2.2 Let S be a regular semigroup with an E-inverse transversal S0. Then (1) xx0y0y ∈ S(y0y, xx0) for any x, y ∈ S; (2) y0x0xy ∈ E(S) for any x, y ∈ S; (3) (x0x)0, (yy0)0 ∈ E(S0) for any x, y ∈ S. Proof. (1) Let x, y ∈ S, and let g = xx0y0y. Then gy0y = xx0y0yy0y = xx0y0y = g, xx0g = xx0xx0y0y = xx0y0y = g, y0ygxx0 = y0yxx0y0yxx0 = y0yxx0, so xx0y0y ∈ S(y0y, xx0). (2) Let x, y ∈ S, then y0x0xyy0x0xy = y0x0xyy0y = y0x0xy, thus y0x0xy ∈ E(S). (3) Let x, y ∈ S, then x0x, yy0 ∈ E(S), by Lemma 2.1, we have (x0x)0, (yy0)0 ∈ E(S0). 740 Yuxiao Gao and Yingchun Wang Lemma 2.3 Let S be a regular semigroup with an inverse transversal S0. Then the following conditions are equivalent: (1) S is orthodox; (2) S0 is a strongly E-inverse transversal of S. Proof. (1) ⇒ (2) Suppose that S is orthodox , and that for any x, y ∈ S x0xx0x = x0x ∈ E(S),yy0yy0 = yy0 ∈ E(S), then x0xyy0,xx0y0y ∈ E(S), thus S0 is a strongly E-inverse transversal of S. (2) ⇒ (1) Suppose that S0 is a strongly E-inverse transversal of S, hence x0xyy0, xx0y0y ∈ E(S) for any x, y ∈ S, moreover xyy0x0xy = xx0xyy0x0xyy0y = xx0xyy0y = xy, y0x0xyy0x0 = y0yy0x0xyy0x0xx0 = y0yy0x0xx0 = y0x0. We deduce that (xy)0 = y0x0, by Lemma 1.1, hence S is orthodox. Then we have the following result: Theorem 2.4 Let S be a regular semigroup. Then the following statements are equivalent: (1) S0 is a strongly E-inverse transversal of S; (2) S is orthodox; (3) fe ∈ S(e, f) for any e, f ∈ E(S); (4) V (y)V (x) ⊆ V (xy) for any x, y ∈ S; (5) V (e) ⊆ E(S) for any e ∈ E(S). Proof. By Lemma 2.3, It is clear that (1) ⇔ (2). (2) ⇒ (3). Suppose that S is orthodox, let e, f ∈ E(S), and let g = fe. then ge = eg = g, egf =(ef)2 = ef. and so g = fe ∈ S(e, f). (3) ⇒ (4). Let x, y ∈ S and let x ∈ V (x),y ∈ V (y). Write xx = e, yy = f, and let g ∈ S(e, f).Then (xy)(ygx)(xy)= fgey = xgy = xxxgyyy = x(egf)y = x(ef)y = xxxyyy = xy, (ygx)(xy)(ygx)= ygefgx = yg2x = ygx. Regular semigroups with strongly E-inverse transversals 741 and so ygx ∈ V (xy) for all g in S(xx, yy). From (3) it thus follows that yx = y(yyxx)x ∈ V (xy), exactly as required. (4) ⇒ (5). Let e ∈ E(S) and x be an inverse of e: xex = x, exe = e. Now xe and xe are both idempotent,and so each is an inverse of itself. By (4) we deduce that (ex)(xe) is an inverse of (xe)(ex), that is to say, that ex2e is an inverse of xe2x = xex = x. Hence x = x(exe)x =(xex)(xex)=(xex)2 = x2, and so x is an idempotent as required. (5) ⇒ (2). Let e, f ∈ E(S). There exists an idempotent g in V (ef) (an element of sandwich set S(e, f)). But then ef, being an inverse of the idempotent g, must itself be an idempotent. Hence S is orthodox. By lemma 1.2, we cannot expect the set V (e) of an arbitrary element of S to be solely determined by properties of E(S), but in fact the properties of E(S) are highly influential, in the sense that if we know a single inverse x of x then V (x) is wholly determined by x and by E(S). Lemma 2.5 Let S be an orthodox semigroup, for any x ∈ S,ifx is an inverse of x. Then E E V (x)=Jxxx Jxx . E E Proof. Let e ∈ Jxx and f ∈ Jxx . Then xxexx = xx, xxfxx = xx, and so x(exf)x = xxexxxxxfxxx = x(xxexx)x(xxfxx)x = xxxxxxx = x, and (exf)x(exf)= exxxfxxxxxexxxf = ex(xxfxx)x(xxexx)xf = ex(xx)x(xx)xf = exf. E E Thus Jxxx Jxx ⊆ V (x). 742 Yuxiao Gao and Yingchun Wang Conversely, suppose that x∗ ∈ V (x). Then x∗ = x∗xx∗ = x∗xxxx∗. Now, from (x∗x)(xx)(x∗x)=x∗(xxx)x∗x = x∗xx∗x = x∗x and (xx)(x∗x)(xx)=x(xx∗x)xx = xxxx = xx ∗ E ∗ E we deduce that x x ∈ Jxx. A similar argument shows that xx ∈ Jxx , and it E E is now that V (x) ⊆ Jxxx Jxx . Then we have the following theorem: Theorem 2.6 Let S be a regular semigroup with a strongly E-inverse transver- sal S0. Then for any e ∈ E(S) the equivalence relation μ = {(x, y) ∈ S × S | x0ex = y0ey, xex0 = yey0} is the greatest idempotent separating congruence on S.
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