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.