Abstract The class of identical inclusions was defined by E. S. Lyapin. This class of nonelementary universal formulas is situated strictly between identities and universal positive formulas. These formulas can be written as identical equalities of subsets of a free algebra, in particular, of $$X^+$$ X + or of a free unary semigroup. Classes of semigroups defined by identical inclusions are called inclusive varieties . We describe the lattice of inclusive varieties of Clifford semigroups modulo the lattice of inclusive varieties of groups and show that even in the lowest layers of the lattice of inclusive varieties of Abelian groups there exist continual parts. We characterize Clifford semigroups defined by their identical inclusions up to isomorphism. A partial classification of nonperiodic and of reducible Abelian groups up to inclusive equivalence and a list of groups from this class definable by its identical inclusions up to isomorphism are obtained. We consider a syntactic presentation of all inclusive varieties of Clifford semigroups over groups from an inclusive variety of groups.

Abstract The commuting graph of a finite non-commutative semigroup S is the simple graph whose vertices are the non-central elements of S , and two distinct vertices a and b are adjacent if and only if $$ab=ba$$ a b = b a . In this paper we investigate commuting graphs of finite non-commutative Rees matrix semigroups over semigroups.

Abstract For a locally compact group $$G$$ G , let $$ AP (G)$$ A P ( G ) and $$ WAP (G)$$ W A P ( G ) be respectively the $$C^{*}$$ C ∗ -algebras of almost periodic and weakly almost periodic functions on $$G$$ G . For a bounded continuous function $$f$$ f on $$G$$ G , $$f$$ f is said to be strictly w.a.p. if its double orbit $$O(f)$$ O ( f ) is relatively weakly compact and $$f$$ f is said to be strictly uniformly continuous if its double orbit is uniformly equicontinuous on $$G$$ G . The $$C^{*}$$ C ∗ -algebras of such functions are denoted, respectively, by $$\textit{WS}(G)$$ WS ( G ) and $$ UCS (G)$$ U C S ( G ) . Then $$\textit{WS}(G) \subset UCS (G)$$ WS ( G ) ⊂ U C S ( G ) and $$ AP (G) \subset \textit{WS}(G) \subset WAP (G)$$ A P ( G ) ⊂ WS ( G ) ⊂ W A P ( G ) . $$G$$ G is called a $$ WS $$ WS -group if $$\textit{WS}(G) = WAP (G)$$ WS ( G ) = W A P ( G ) . We will show that if a discrete FC -group $$G$$ G is a $$ WS $$ WS -group, then its center is of finite index in $$G$$ G . A noncompact locally compact group $$G$$ G is minimally w.a.p., if $$ WAP (G) = AP (G) \oplus C_{0}(G)$$ W A P ( G ) = A P ( G ) ⊕ C 0 ( G ) . If $$G$$ G is minimally w.a.p., then $$\textit{WS}(G) = AP (G)$$ WS ( G ) = A P ( G ) , i.e., if the double orbit of a bounded continuous function $$f$$ f is relatively weakly compact then it is relatively norm compact. It is known that for $$n \ge 2$$ n ≥ 2 , the motion group $$M(n)$$ M ( n ) , and the special linear group $$\textrm{SL}(n,\,\mathbb {R})$$ SL ( n , R ) are minimally w.a.p. On the other hand, there exist locally compact groups $$G$$ G such that $$\textit{WS}(G) = AP (G)$$ WS ( G ) = A P ( G ) but $$G$$ G is not minimally w.a.p. We will show that if $$G$$ G is an IN -group and $$K = K_{G}$$ K = K G is the intersection of all closed invariant neighborhoods of the identity of $$G$$ G , then $$ UCS (G) = UCS (G/K)$$ U C S ( G ) = U C S ( G / K ) and $$\textit{WS}(G) = \textit{WS}(G/K)$$ WS ( G ) = WS ( G / K ) . We will identify the strictly w.a.p. functions on the $$ax + b$$ a x + b group. We will also show that $$ UCS (\textrm{SL}(2,\,\mathbb {R}))$$ U C S ( SL ( 2 , R ) ) only contains the constant functions.

Abstract We describe the structure of E -dense acts over E -dense semigroups in an analogous way to that for inverse semigroup acts over inverse semigroups. This is based, to a large extent, on the work of Schein on representations of inverse semigroups by partial one-to-one maps. We also study cancellative actions of semigroups as a type of generalisation of group actions and characterise locally free cancellative acts over E -dense semigroups that are also E -unitary.