On Automatic Rees Matrix Semigroups

We consider a Rees matrix semigroup S = M[U; I, J; P] over a semigroup U, with I and J finite index sets, and relate the automaticity of S with the automaticity of U. We prove that if U is an automatic semigroup and S is finitely generated then S is an automatic semigroup. If S is an automatic semigroup and there is an entry p in the matrix P such that pU ¹ = U then U is automatic. We also prove that if S is a prefix-automatic semigroup, then U is a prefix-automatic semigroup.     

A Structure Preserving Embedding of Semigroups Into Regular Semigroups

For any semigroup S a regular semigroup 𝒞(S) that embeds S can be constructed as the direct limit of a sequence of semigroups each of which contains a copy of its predecessor as a subsemigroup whose elements are regular. The construction is modified here to obtain an embedding of S into a regular semigroup R such that the nontrivial maximal subgroups of R are isomorphic to the Schützenberger groups of S and such that the restriction to S of any of Green's relations on R is the corresponding Green's relation on S.     

E-special Ordered Regular Semigroups

An ordered regular semigroup S is E-special if for every x ∈ S there is a biggest x ⁺ ∈ S such that both xx ⁺ and x ⁺ x are idempotent. Every regular strong Dubreil–Jacotin semigroup is E-special, as is every ordered completely simple semigroup with biggest inverses. In an E-special ordered regular semigroup S in which the unary operation x → x ⁺ is antitone the subset P of perfect elements is a regular ideal, the biggest inverses in which form an inverse transversal of P if and only if S has a biggest idempotent. If S ⁺ is a subsemigroup and S does not have a biggest idempotent, then P contains a copy of the crown bootlace semigroup.     

Projective S-acts and Exact Functors

Let S be a semigroup. In this paper, projective S-acts and exact sequences in S-Act are studied. It is shown that, for a unitary S-act P, the functor Hom(P, –) is exact if and only if P Se for some idempotent e S.The research was supported by the National Natural Science Foundation of China.1991 Mathematics Subject Classification: 20M50     

On the growth of generating sets for direct powers of semigroups

For a semigroup S its d-sequence is d(S)=(d ₁,d ₂,d ₃,…), where d _i is the smallest number of elements needed to generate the ith direct power of S. In this paper we present a number of facts concerning the type of growth d(S) can have when S is an infinite semigroup, comparing them with the corresponding known facts for infinite groups, and also for finite groups and semigroups.     

Apery Sets of Numerical Semigroups #

Let S be a numerical semigroup. We examine a particular subset of the Apery set of S and establish a correspondence between this subset and the holes of S . This correspondence allows us to establish conditions for S to be almost symmetric.     

Characterizing compact Clifford semigroups that embed into convolution and functor-semigroups

We study algebraic and topological properties of the convolution semigroup of probability measures on a topological groups and show that a compact Clifford topological semigroup S embeds into the convolution semigroup P(G) over some topological group G if and only if S embeds into the semigroup exp(G)\exp(G) of compact subsets of G if and only if S is an inverse semigroup and has zero-dimensional maximal semilattice. We also show that such a Clifford semigroup S embeds into the functor-semigroup F(G) over a suitable compact topological group G for each weakly normal monadic functor F in the category of compacta such that F(G) contains a G-invariant element (which is an analogue of the Haar measure on G).     

Globals of Idempotent Semigroups

If S is a semigroup, the global (or the power semigroup) of S is the set P(S) of all nonempty subsets of S equipped with a naturally defined multiplication. A class K of semigroups is globally determined if any two semigroups of K with isomorphic globals are themselves isomorphic. We study properties of globals of idempotent semigroups and show, in particular, that the class of normal bands is globally determined.     

On the Buchsbaumness of the Associated Graded Ring of a One-Dimensional Local Ring

Let (R, m) be a Noetherian, one-dimensional, local ring, with |R/m|=∞. We study when its associated graded ring G(m) is Buchsbaum; in particular, we give a theoretical characterization for G(m) to be Buchsbaum not Cohen–Macaulay. Finally, we consider the particular case of R being the semigroup ring associated to a numerical semigroup S: we introduce some invariants of S, and we use them in order to give a necessary and a sufficient condition for G(m) to be Buchsbaum.     

Two-sided networks for completely simple semigroups

Let S be a completely simple semigroup represented as a Rees matrix semigroup M(I,G,P) with normalized sandwich matrix P. On the congruence lattice C(S) of S we consider the relations T _i, K and T _r which identify congruences with the same left trace, kernel and right trace, respectively. These are equivalences whose classes are intervals. The upper and lower ends of these intervals induce the following operators on C(S) T_l, K, T_r, t_l, k and t_r .We construct here the semigroup generated by these operators as a homomorphic image of a semigroup given by generators and relations and demonstrate the minimality of the latter.     

Ordered semigroups having the <Emphasis Type="Italic">P</Emphasis>-property

In this paper we obtain the following main results. The ordered semigroups which have the P-property are decomposable into archimedean semigroups. Moreover, they are decomposable into semigroupswith the P-property. Conversely, if an ordered semigroup S is a complete semilattice of semigroups which have the P-property, then S itself also has the P-property. An ordered semigroup is CS-indecomposable and has the P-property if and only if it is archimedean. If S is an ordered semigroup, then the relation N:= {(a, b) | N(a) = N(b)} (here N(a) is a filter of S generated by a (a ∈ S)) is the least complete semilattice congruence on S and the class (a) _N is a CS-indecomposable subsemigroup of S for each a ∈ S. We introduce the notion of the P _m -property and describe it in terms of the P-property. Our approach simplifies the proofs of the corresponding results about unordered semigroups. The text was submitted by the authors in English.     

Congruences on eventually regular semigroups

Let S be an eventually regular semigroup. The extensively P-partial congruence pairs and P-partial congruence pairs for S are defined. Furthermore, the relationships between the lattice of congruences on S, the lattice of P-partial kernel normal systems for S, the lattice of extensively P-partial kernel normal systems for S and the poset of P-partial congruence pairs for S are explored.     

On finiteness conditions for Rees matrix semigroups

Let T=[S; I; J; P] be a Rees matrix semigroup where S is a semigroup, I and J are index sets, and P is a J × I matrix with entries from S, and let U be the ideal generated by all the entries of P. If U has finite index in S, then we prove that T is periodic (locally finite) if and only if S is periodic (locally finite). Moreover, residual finiteness and having solvable word problem are investigated.     

Semiprimitivity of Orthodox Semigroup Algebras

Let S be a finite orthodox semigroup or an orthodox semigroup where the idempotent band E(S) is locally pseudofinite. In this paper, by using principal factors and Rukolaǐne idempotents, we show that the contracted semigroup algebra R₀[S] is semiprimitive if and only if S is an inverse semigroup and R[G] is semiprimitive for each maximal subgroup G of S. This theorem strengthens previous results about the semiprimitivity of inverse semigroup algebras.     

ON THE STRUCTURE OF THE JACOBSON RADICAL OF GRADED RINGS

Abstract

We introduce a new large class of semigroups S including all locally finite, completely regular and strongly π-regular linear semigroups. For any semigroup S in the class and any S-graded ring R, the structure of the Jacobson radical of R is reduced to the radicals of subrings graded by the maximal subgroups of S. Many results on radicals follow from this reduction in a unified way. In two special cases the reduction is simplified.     

The value semiring of an algebroid curve

Abstract

We study the value semiring Γ, equipped with the tropical operations, associated to an algebroid curve. As a set, Γ determines and is determined by the well-known value semigroup S and we prove that Γ is always finitely generated in contrast to S. In particular, for a plane curve, we present a straightforward way to obtain Γ in terms of the semiring (or the semigroup) of each branch of the curve and the mutual intersection multiplicity of its branches. In the analytic case, this allows us to relate the results of Zariski and Waldi that characterize the topological type of the curve.     

On character amenability of semigroup algebras

We study the character amenability of semigroup algebras. We work on general semigroups and certain semigroups such as inverse semigroups with a finite number of idempotents, inverse semigroups with uniformly locally finite idempotent set, Brandt and Rees semigroup and study the character amenability of the semigroup algebra l¹(S) in relation to the structures of the semigroup S. In particular, we show that for any semigroup S, if ?¹(S) is character amenable, then S is amenable and regular. We also show that the left character amenability of the semigroup algebra ?¹(S) on a Brandt semigroup S over a group G with index set J is equivalent to the amenability of G and J being finite. Finally, we show that for a Rees semigroup S with a zero over the group G, the left character amenability of ?¹(S) is equivalent to its amenability, this is in turn equivalent to G being amenable.     

Transition Semigroups of Banach Space-Valued Ornstein–Uhlenbeck Processes

We investigate the transition semigroup of the solution to a stochastic evolution equation dX(t)=AX(t)dt+dW _H (t), t0, where A is the generator of a C ₀-semigroup S on a separable real Banach space E and W _H (t) _t0 is cylindrical white noise with values in a real Hilbert space H which is continuously embedded in E. Various properties of these semigroups, such as the strong Feller property, the spectral gap property, and analyticity, are characterized in terms of the behaviour of S in H. In particular we investigate the interplay between analyticity of the transition semigroup, S-invariance of H, and analyticity of the restricted semigroup S _H .     

Retracts and monomorphism monoids in semigroup varieties

If a semigroup varietyV contains the variety of commutative three-nilpotent semigroups, or if it is a variety of bands containing all semilattices, then, for anyA∈V and any left cancellative monoidM, there is a semigroupS∈V such thatA is a retract ofS andM is isomorphic to the monoid of all injective endomorphisms ofS.