Open Math. 2015; 13: 931–939 Open Mathematics Open Access Research Article Ze Gu and Xilin Tang* On V n-semigroups DOI 10.1515/math-2015-0085 Received November 18, 2014; accepted December 3, 2015. Abstract: In this paper, we give some new characterizations of orthodox semigroups in terms of the set of inverses of idempotents. As a generalization, a new class of regular semigroups, namely V n-semigroups, is introduced. Also, we give a characterization of V n-semigroups and investigate some properties of V n-semigroups. Furthermore, we show that the class of V n-semigroups is closed under direct products and homomorphic images. However, regular subsemigroups of V n-semigroups (n 2) are not necessarily V n-semigroups in general. Therefore, the class of V n-semigroups (n 2) does not form an e-variety. Finally, we obtain that a E-solid semigroup S is a V 2-semigroup if and only if S is orthodox. Keywords: Orthodox semigroups, V n-semigroups, V 2-semigroups, E-solid semigroup; e-variety MSC: 20M10 1 Introduction and preliminaries Let S be a semigroup. An element a of S is called regular if there exists x S such that a axa. S is called 2 D regular if all its elements are regular. Regular semigroups are important research objects in semigroup theory. An element e S is an idempotent if e2 e. We denote by E the set of idempotents of S. The subsemigroup of S 2 D generated by E, denoted by E , consists of all finite products of elements in E. A semigroup S is called a band if h i all the elements of S are idempotents. A commutative band is called a semilattice. We say that a S is an inverse 0 2 of a S if a aa a and a a aa . A semigroup S is called an inverse semigroup if every element of S has a 2 D 0 0 D 0 0 unique inverse. From Theorem 5.1.1 in [1], we know that S is an inverse semigroup if and only if S is regular and E is a semilattice. A regular semigroup S is called orthodox if E is a band. An element a of a semigroup S is called completely regular if there exists x S such that a axa and ax xa. The semigroup S is called completely 2 D D regular if all the elements of S are completely regular. A regular semigroup S is called E-solid (quasi-orthodox) if E is completely regular. A completely regular and orthodox semigroup is called a orthogroup. A semigroup h i is called simple if it has no proper ideal. A completely regular and simple semigroup is called completely simple. A completely simple and orthodox semigroup is called a rectangular group. Results on these semigroups can be seen in [2–6]. Let S be a regular semigroup. An inverse subsemigroup T of S is called an inverse transversal of S if T contains a unique inverse of x for every x S. The class of regular semigroups with inverse transversals was 2 introduced by Blyth and McFadden in [7]. Necessary and sufficient conditions for an inverse subsemigroup to be an inverse transversal were given by the second author of this paper in [8]. Related work on regular semigroups with inverse transversals can be seen in [9–13]. Ze Gu: Department of Mathematics, South China University of Technology, Guangzhou, Guangdong, 510640, P.R. China, E-mail: [email protected] *Corresponding Author: Xilin Tang: Department of Mathematics, South China University of Technology, Guangzhou, Guangdong, 510640, P.R. China, E-mail: [email protected] © 2015 Gu and Tang, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. 932 Z. Gu, X. Tang A semigroup with an identity is called a monoid. If a semigroup S is not a monoid, we may adjoin a new element 1 to S with the property 1a a1 a for all a S 1 . Then S 1 is a monoid. We now define D D 2 [ f g [ f g ( S if S has an identity; S 1 D S 1 otherwise: [ f g Green’s relations are five equivalences on a semigroup. They are important tools in the description and decomposition of semigroups. Let S be a semigroup. We define the following relations on S: aLb S 1a S 1b , D I aRb aS 1 bS 1 , D I aJ b S 1aS 1 S 1bS 1 , D I H L R D \ I D L R: D _ From Proposition 2.1.3 in [1], L and R commute. Thus D L R R L. The five equivalence relations are D ı D ı called Green’s relations. We denote by La (Ra;Ja;Ha;Da) the equivalence L-class (R-class, J -class, H-class, D-class) containing the element a. We define LE L .E E/ and RE R .E E/. D \ D \ Define a relation V on a regular semigroup S as V .a; b/ S S aba a; bab b . We denote by V .a/ D f 2 W D D g the set b S .a; b/ V . Then V .a/ is the set of inverses of a. Denote V .A/ S V .a/ for any non-empty f 2 W 2 g D a A subset A of S. For any positive integer n, we denote by Vn the n times composition of2 the relation V. In particular, V0 is the equality relation on S. Similarly, we denote V n.a/ b S .a; b/ Vn and V n.A/ S V n.a/ D f 2 W 2 g D a A respectively. 2 If S is a regular semigroup with an inverse transversal T , then V .T / S. If S is just a regular semigroup, we D want to know when T is an inverse transversal of V .T / for any inverse subsemigroup T of S. In Section 2, we find that orthodox semigroups satisfy the condition and give some new characterizations of orthodox semigroups. It is well known from [4] that a regular semigroup S is an inverse semigroup if and only if every e E has 2 a unique inverse. Similarly, we give some characterizations of orthodox semigroups in Section 3. Furthermore, we investigate the relationship between the characterizations and the results given in Section 2. As a generalization of inverse semigroups and orthodox semigroups, we introduce a new class of regular semigroups, called V n-semigroups, in Section 4. Also, we give a characterization of V n-semigroups and investigate some properties of V n-semigroups. Moreover, we show that a semilattice of V n-semigroups, the direct products of V n-semigroups, the homomorphic images of V n-semigroups and the ideal extensions of V n-semigroups by V n- semigroups are still V n-semigroups. However, regular subsemigroups of V n-semigroups (n 2) are not necessarily V n-semigroups in general. Therefore, the class of V n-semigroups (n 2) does not form an e-variety. Finally, we obtain that a E-solid semigroup S is a V 2-semigroup if and only if S is orthodox. We list the following five results that will be used in our paper. Result 1.1 (Theorem 2.3.4 (2) and Proposition 2.3.7 [1]). Let a; b be elements of a semigroup S. (1) If Ra Lb and La Rb contain idempotents e; f respectively, then Hb contains an inverse a of a such that \ \ aa e; a a f . D D (2) ab Ra Lb if and only if La Rb contains an idempotent. 2 \ \ Result 1.2 (Theorem [14]). Let S be a regular semigroup with set E of idempotents. Then for any positive integer n, V n.E/ V .En/ En 1: D D C Result 1.3 (Theorem 6.2.1 and Theorem 6.2.4 [1]). Let S be a regular semigroup with set E of idempotents. Then the following conditions are equivalent: (1) S is orthodox; (2) . a; b S/ V .b/V .a/ V .ab/; 8 2 Â (3) . e E/ V .e/ E; 8 2 Â On V n-semigroups 933 (4) . a; b S/ V .a/ V .b/ V .a/ V .b/. 8 2 \ ¤ ; ) D Result 1.4 (Theorem 3 [2]). Let S be a regular semigroup with set E of idempotents. Then S is a E-solid semigroup if and only if RE LE LE RE . ı D ı Result 1.5 (Lemma 1 [15]). Let be a congruence on a regular semigroup S and let A; B be any elements of S= such that AVB in S=. Then there exist elements a; b in S such that a A; b B and aVb in S. 2 2 2 V.e/V.e/ V.e/ D For a regular semigroup S, we have the following result. Lemma 2.1. Let S be a regular semigroup with set E of idempotents and IS be the set of inverse subsemigroups of S. Then for any positive integer n, (1) . e E/V n.e/ V n.e/V n.e/; 8 2 Â (2) . T IS/V n.T / V n.T /V n.T /. 8 2 Â Proof. (1). If n 1, then aeVe for any a V .e/. Thus a .ae/a V .e/V .e/. We next prove the case of D n 2 D 2 n > 1. For any a V .e/, there exist e1; e2; : : : ; en 1 S such that aVe1Ve2V Ven 1Ve.