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Semigroups with Operation-Compatible Green’S Quasiorders View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Essex Research Repository SEMIGROUPS WITH OPERATION-COMPATIBLE GREEN'S QUASIORDERS ZSOFIA´ JUHASZ,´ ALEXEI VERNITSKI Abstract. We call a semigroup on which the Green's quasiorder ≤J (≤L, ≤R) is operation-compatible, a ≤J -compatible (≤L-compatible, ≤R-compatible) semigroup. We study the classes of ≤J -compatible, ≤L-compatible and ≤R-compatible semigroups, using the smallest operation-compatible quasiorders containing Green's quasiorders as a tool. We prove a number of results, including the following. The class of ≤L-compatible (≤R-compatible) semigroups is closed under taking homomorphic images. A regular periodic semigroup is ≤J -compatible if and only if it is a semilattice of simple semi- groups. Every negatively orderable semigroup can be embedded into a negatively orderable ≤J -compatible semigroup. 1. Introduction Green's relations L, R and J are one of the most important tools in studying the structure of semigroups. They can also be viewed from a less common angle: as being defined via quasiorders (or pre- orders), which we shall refer to as Green's quasiorders and denote by ≤L, ≤R and ≤J , respectively. Studying the properties of these qua- siorders is of interest, because of the importance of Green's relations and due to the fact that in a certain sense these associated quasiorders contain `more information' about a semigroup than Green's relations: given only a Green's quasiorder on a semigroup we can reconstruct the corresponding Green's relation, whereas the converse is not true. We shall call a semigroup S ≤L-compatible, ≤R-compatible and ≤J - compatible, respectively, if ≤L, ≤R and ≤J is operation-compatible on S. The aim of this paper is to explore some properties of the classes of ≤L-compatible, ≤R-compatible and ≤J -compatible semigroups. These classes are natural to consider; operation-compatible quasiorders have the convenient property that the equivalences induced by them are con- gruences, hence yield factor semigroups. We shall denote the smallest operation-compatible quasiorders containing ≤ , ≤ and ≤ by ≤ ◦ , L R J J 1 2 ZSOFIA´ JUHASZ,´ ALEXEI VERNITSKI ≤ ◦ and ≤ ◦ , respectively. In [9] it was shown that there is a close con- R J nection between ≤ ◦ and the filters of a semigroup, and thus ≤ ◦ can J J be used to determine the lattice of filters and the largest semilattice image of a semigroup. 2. Definitions and observations 2.1. Main concepts. A quasiorder (or preorder) on a set is a reflexive and transitive relation. If S is a semigroup, by S1 one denotes S if it has an identity element or, otherwise, S with an added identity element. We shall call Green's quasiorders the relations defined on every semigroup as follows: Definition 2.1. For any elements s; t of a semigroup S let 1 • s ≤L t if and only if s = xt for some x 2 S , 1 • s ≤R t if and only if s = ty for some y 2 S , 1 • s ≤J t if and only if s = xty for some x; y 2 S . It is easy to show that the relations ≤L; ≤R and ≤J are quasiorders −1 −1 −1 and that L = ≤L \ ≤L , R = ≤R \ ≤R and J = ≤J \ ≤J . A quasiorder ≤ on a semigroup S is left (right) operation-compatible if for all a; b; c 2 S, a ≤ b implies ca ≤ cb (ac ≤ bc). A quasiorder is operation-compatible if it is both left and right operation-compatible. Clearly, ≤L (≤R) is right (left) operation-compatible on every semi- group. However, Green's quasiorders are not operation-compatible in general. As operation-compatible quasiorders on any semigroup form a complete lattice, for any quasiorder on a semigroup there exists a smallest operation-compatible quasiorder containing it. Definition 2.2. We call a semigroup ≤J -compatible (≤L-compatible, ≤R-compatible) if ≤J (≤L, ≤R) is operation-compatible on S. Definition 2.3. Denote by ≤ ◦ (≤ ◦, ≤ ◦ ) the smallest operation-compatible J L R quasiorder containing ≤J (≤L, ≤R). Relations ≤ ◦ , ≤ ◦ and ≤ ◦ will be a useful instrument for us in this pa- J L R per because, obviously, a semigroup is ≤J -compatible (≤L-compatible, ≤ -compatible) if and only if ≤ = ≤ ◦ (≤ = ≤ ◦, ≤ = ≤ ◦ ). R J J L L R R In [10] a description of ≤ ◦ , ≤ ◦ and ≤ ◦ was given. In Lemma 2.1 J J R below we give another description, which will be convenient later. In this lemma, for any relation θ, θ denotes the transitive closure of θ. SEMIGROUPS WITH OPERATION-COMPATIBLE GREEN'S QUASIORDERS 3 Let S be a semigroup. Define the relation ≺ ◦ as follows: for any J s; t 2 S let s ≺ ◦ t if and only if s = t s t and t = t t for some J 1 1 2 1 2 1 t ; t ; s 2 S . Define the relation ≺ ◦ (≺ ◦ ) as follows: for any s; t 2 S 1 2 1 L R 1 1 let s ≺ ◦ t (s ≺ ◦ t) if and only if t 2 S, t ; s 2 S (t 2 S, t ; s 2 S ). L R 2 1 1 1 2 1 Lemma 2.1. In every semigroup (1) ≤ ◦ = ≺ ◦ J J (2) ≤ ◦ = ≺ ◦ L L (3) ≤ ◦ = ≺ ◦ . R R Proof. We only prove Statement 1, since Statements 2 and 3 can be verified similarly. 1 If a ≤ b then a = sbt for some s; t 2 S ; since a = sbt ≺ ◦ sb ≺ ◦ b, J J J we have a≺ ◦ b. Therefore, ≤ ⊆ ≺ ◦ . It is obvious that if a ≺ ◦ b then J J J J 1 1 for any s; t 2 S , sat ≺ ◦ sbt. Hence, if a≺ ◦ b then for any s; t 2 S J J sat≺ ◦ sbt. Therefore, ≺ ◦ is operation-compatible. Obviously, ≺ ◦ is J J J transitive. Therefore, ≤ ◦ ⊆ ≺ ◦ . J J It is obvious that ≺ ◦ is contained in the operation-compatible closure J of ≤ . Hence, ≺ ◦ is contained in the transitive operation-compatible J J closure of ≤ , which is exactly ≤ ◦ . Therefore, ≤ ◦ ⊇ ≺ ◦ . J J J J 2.2. Examples of classes of ≤J -compatible semigroups. Proposition 2.1. Every group and every commutative semigroup is ≤J -compatible, ≤L-compatible and ≤R-compatible. Proof. The result follows from the fact that in a group or in a commu- tative semigroup ≺ ◦ ⊆ ≤ , ≺ ◦ ⊆ ≤ and ≺ ◦ ⊆ ≤ and from Lemma J J L L R R 2.1. As we shall see in Sections 4 and 5, every band is ≤J -compatible, but not necessarily ≤L-compatible and ≤R-compatible. 2.3. Monoids. As the following statements demonstrate, results con- cerning ≤ ◦ are not affected by a semigroup being a monoid; however, J results concerning ≤ ◦ and ≤ ◦ are affected by this fact. L R 4 ZSOFIA´ JUHASZ,´ ALEXEI VERNITSKI Proposition 2.2. Consider a semigroup S and a monoid M = S [ 1 with the neutral element 1, where 1 62 S. Then the relation ≤ ◦ on S J is equal to the restriction of ≤ ◦ on S. J Proof. This follows from the description of ≤ ◦ in Lemma 2.1. J Proposition 2.3. In every monoid ≤ ◦ = ≤ ◦ = ≤ ◦ . J L R Proof. From the definition it follows that in any monoid M we have ≺ ◦ = ≺ ◦ = ≺ ◦ . Therefore, by Lemma 2.1, ≤ ◦ = ≤ ◦ = ≤ ◦ . J L R J L R 3. Congruences 3.1. Induced equivalence relations. For any element s in a semi- group S and any congruence θ on S, sθ shall denote the image of s under the natural homomorphism S ! S/θ. Lemma 3.1. Let S be a semigroup and let s; t 2 S be such that s ≤ ◦ t J θ θ θ θ (s ≤ ◦ t, s ≤ ◦ t). Then for any congruence θ on S, s ≤ ◦ t (s ≤ ◦ t , L R J L θ θ s ≤ ◦ t ) in S/θ. R Proof. If s ≤ ◦ t then by Lemma 2.1 there exist s = s ; s ; : : : ; s = J 0 1 n t 2 S such that s ≺ ◦ s for every 0 ≤ i ≤ n − 1. Fix an arbitrary i J i+1 1 0 ≤ i ≤ n−1. Then si = abc and si+1 = ac for some a; b; c 2 S . Hence θ θ θ θ θ θ θ 1 θ 1 si = a b c and si+1 = a c (where for 1S 2 S we have 1S = 1T 2 T ), θ θ θ θ and so s ≺ ◦ s . Therefore by Lemma 2.1 s ≤ ◦ t . (The proof is i J i+1 J similar for ≤ ◦ and ≤ ◦ .) L R ◦ ◦ ◦ −1 Definition 3.1. Denote by J , L and R the equivalences ≤ ◦ \ ≤ ◦ , J J −1 −1 ≤ ◦ \ ≤ ◦ and ≤ ◦ \ ≤ ◦ , respectively. L L R R For any operation-compatible quasiorder ≤, ≤ \ ≤−1 is a congruence ◦ ◦ ◦ (see [13] for instance), hence J , L and R are congruences. ◦ ◦ Definition 3.2. Let us say that a semigroup is J -trivial (L-trivial, ◦ ◦ ◦ ◦ R-trivial) if J (L, R) is the identity relation on S. We call a quasiorder on a semigroup S a negative quasiorder if st ≤ s and st ≤ t for every s; t in S; S is called negatively orderable if there exists an operation-compatible negative partial order on S. SEMIGROUPS WITH OPERATION-COMPATIBLE GREEN'S QUASIORDERS 5 ◦ Proposition 3.1. A semigroup is J -trivial if and only if it is negatively orderable. ◦ Proof. If a semigroup S is J -trivial then, obviously, ≤J is an operation- compatible negative partial order on S. If there is an operation-compatible negative partial order ≤ on S then ≺ ◦ ⊆ ≤, by the definition of ≺ ◦ , J J ◦ hence, ≤ ◦ ⊆ ≤, therefore, ≤ ◦ is an order and, hence, J is the identity J J relation. According to the usual convention, let us call a congruence θ on a ◦ ◦ ◦ semigroup S a J -trivial congruence (L-trivial congruence, R-trivial ◦ ◦ ◦ congruence) if S/θ is a J -trivial semigroup (L-trivial semigroup, R- trivial semigroup).
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