Left Cancellative Semigroup Research Articles

On Pseudo‐contractibility and Ultra Central Approximate Identity of Some Semigroup Algebras

In this paper, we investigate the existence of an ultra central approximate identity for a Banach algebra A and its second dual A∗∗. Also, we prove that for a left cancellative regular semigroup S, ℓ1(S)∗∗ has an ultra central approximate identity if and only if S is a group. As an application, we show that for a left cancellative semigroup S, ℓ1(S)∗∗ is pseudo‐contractible if and only if S is a finite group. We also study this property for φ‐Lau product Banach algebras and the module extension Banach algebras.

Topological Grading of Semigroup C*-Algebras

The paper deals with the abelian cancellative semigroups and the reduced semigroup C*-algebras. It is supposed that there exist epimorphisms from the semigroups onto the group of integers modulo n. For these semigroups we study the structure of the reduced semigroup C*-algebras which are also called the Toeplitz algebras. Such a C*-algebra can be defined for any non-abelian left cancellative semigroup. It is a very natural object in the category of C*-algebras because this algebra is generated by the left regular representation of a semigroup. In the paper, by a given epimorphism σ we construct the grading of a semigroup C*-algebra. To this aim the notion of the σ-index of a monomial is introduced. This notion is the main tool in the construction of the grading. We make use of the σ-index to define the linear independent closed subspaces in the semigroup C*-algebra. These subspaces constitute the C*-algebraic bundle, or the Fell bundle, over the group of integers modulo n. Moreover, it is shown that this grading of the reduced semigroup C*-algebra is topological. As a corollary, we obtain the existence of the contractive linear operators that are non-commutative analogs of the Fourier coefficients. Using these operators, we prove the result on the geometry of the underlying Banach space of the semigroup C*-algebra

An inverse semigroup approach to the C*-algebras and crossed products of cancellative semigroups

We give a newdefinition of the semigroup C*-algebra of a left cancellative semigroup, which resolves problems of the construction by X. Li.Namely, the newconstruction is functorial, and the independence of ideals in the semigroup does not influence the independence of the generators. It has a group C*-algebra as a natural quotient. The C*-algebra of the old construction is a quotient of the newone. All this applies both to the full and reduced C*-algebras. The construction is based on the universal inverse semigroup generated by a left cancellative semigroup. We apply this approach to connect amenability of a semigroup to nuclearity of its C*-algebra. Large classes of actions of these semigroups are in one-to-one correspondence, and the crossed products are isomorphic. A crossed product of a left Ore semigroup is isomorphic to the partial crossed product of the generated group.

On Cogrowth, Amenability, and the Spectral Radius of a Random Walk on a Semigroup

Abstract We introduce two natural notions of cogrowth for finitely generated semigroups —one local and one global — and study their relationship with amenability and random walks. We establish the minimal and maximal possible values for cogrowth rates, and show that nonmonogenic-free semigroups are exactly characterised by minimal global cogrowth. We consider the relationship with cogrowth for groups and with amenability of semigroups. We also study the relationship with random walks on finitely generated semigroups, and in particular the spectral radius of the associated Markov operators (when defined) on $\ell _2$-spaces. We show that either of maximal global cogrowth or the weak Følner condition suffices for the spectral radius to be at least $1$; since left amenability implies the weak Følner condition, this represents a generalisation to semigroups of one implication of Kesten’s Theorem for groups. By combining with known results about amenability, we are able to establish a number of new sufficient conditions for (left or right) amenability in broad classes of semigroups. In particular, maximal local cogrowth left implies amenability in any left reversible semigroup, while maximal global cogrowth (which is a much weaker property) suffices for left amenability in an extremely broad class of semigroups encompassing all inverse semigroups, left reversible left cancellative semigroups and left reversible regular semigroups.

Amenability and geometry of semigroups

We study the connection between amenability, Folner conditions and the geometry of finitely generated semigroups. Using results of Klawe, we show that within an extremely broad class of semigroups (encompassing all groups, left cancellative semigroups, finite semigroups, compact topological semigroups, inverse semigroups, regular semigroups, commutative semigroups and semigroups with a left, right or two-sided zero element), left amenability coincides with the strong Folner condition. Within the same class, we show that a finitely generated semigroup of subexponential growth is left amenable if and only if it is left reversible. We show that the (weak) Folner condition is a left quasi-isometry invariant of finitely generated semigroups, and hence that left amenability is a left quasi-isometry invariant of left cancellative semigroups. We also give a new characterisation of the strong Folner condition in terms of the existence of weak Folner sets satisfying a local injectivity condition on the relevant translation action of the semigroup.

Ends of semigroups

We define the notion of the partial order of ends of the Cayley graph of a semigroup. We prove that the structure of the ends of a semigroup is invariant under change of finite generating set and at the same time is inherited by subsemigroups and extensions of finite Rees index. We prove an analogue of Hopf’s Theorem, stating that an infinite group has 1, 2 or infinitely many ends, for left cancellative semigroups and that the cardinality of the set of ends is invariant in subsemigroups and extension of finite Green index in left cancellative semigroups.

On $C^*$-algebras associated to right LCM semigroups

We initiate the study of the internal structure of C*-algebras associated to a left cancellative semigroup in which any two principal right ideals are either disjoint or intersect in another principal right ideal; these are variously called right LCM semigroups or semigroups that satisfy Clifford's condition. Our main findings are results about uniqueness of the full semigroup C*-algebra. We build our analysis upon a rich interaction between the group of units of the semigroup and the family of constructible right ideals. As an application we identify algebraic conditions on S under which C*(S) is purely infinite and simple.

Left Equalizer Simple Semigroups

In this paper we characterize and construct semigroups whose right regular representation is a left cancellative semigroup. These semigroups will be called left equalizer simple semigroups. For a congruence $\varrho$ on a semigroup $S$, let ${\mathbb F}[\varrho]$ denote the ideal of the semigroup algebra ${\mathbb F}[S]$ which determines the kernel of the extended homomorphism of ${\mathbb F}[S]$ onto ${\mathbb F}[S/\varrho]$ induced by the canonical homomorphism of $S$ onto $S/\varrho$. We examine the right colons $({\mathbb F}[\varrho]:_r{\mathbb F}[S])=\{ a\in {\mathbb F}[S]:\ {\mathbb F}[S]a\subseteq {\mathbb F}[\varrho ]\}$ in general, and in that special case when $\varrho$ has the property that the factor semigroup $S/\varrho$ is left equalizer simple.

Primitive program schemata with procedures

Algebraic models of program are widely used for the analysis of program behavior. In this paper we study the equivalence problem for algebraic models of programs with procedures and equivalent transformations of program schemata. Among the algebraic models of programs with procedures we selected a subclass of gateway models and the corresponding primitive program schemata, since in the early papers it was proved that equivalence problem is decidable in this subclass. In this paper we show that in the case when the semantics of basic program statements is specified by decidable balanced left-cancellative semigroup we construct a complete system of equivalent transformations for a particular subclass of primitive program schemata.

Inverse Semigroup C*-Algebras Associated with Left Cancellative Semigroups

AbstractTo each discrete left cancellative semigroup S one may associate an inverse semigroup Il(S), often called the left inverse hull of S. We show how the full and reduced C*-algebras of Il(S) are related to the full and reduced semigroup C*-algebras for S, recently introduced by Li, and give conditions ensuring that these algebras are isomorphic. Our picture provides an enhanced understanding of Li's algebras.

Ascending chain conditions on principal left and right ideals for semidirect products of semigroups

In this paper we investigate the ascending chain conditions on principal left and right ideals for semidirect products of semigroups and show how this is connected to the corresponding problem for rings of skew generalized power series. Let S be a left cancellative semigroup with a unique idempotent e, T a right cancellative semigroup with an idempotent f and \(\omega: T \to \operatorname {End}(S)\) a semigroup homomorphism such that ω(f)=idS. We show that in this case the semidirect product S⋊ωT satisfies the ascending chain condition for principal left ideals (resp. right ideals) if and only if S and T satisfy the ascending chain condition for principal left ideals (resp. right ideals and \(\operatorname {Im}\omega(t)\) is closed for complete inverses for all t∈T). We also give several examples to show that for more general semigroups these implications may not hold.

Semigroup C⁎-algebras and amenability of semigroups

We construct reduced and full semigroup C⁎-algebras for left cancellative semigroups. Our new construction covers particular cases already considered by A. Nica and also Toeplitz algebras attached to rings of integers in number fields due to J. Cuntz. Moreover, we show how (left) amenability of semigroups can be expressed in terms of these semigroup C⁎-algebras in analogy to the group case.

Technique of traces in solving the equivalence problem in algebraic program models

Algebraic models of sequential programs without procedures are considered. The question of applicability of the technique of traces to the solution of the equivalence problem in such models is investigated. Models called balanced left-cancellative semigroups are singled out for which the technique of traces provides an effective decision procedure.

Approximate Amenability of Certain Semigroup Algebras

In the present paper, it is shown that a left cancellative semigroup S (not necessarily with identity) is left amenable whenever the Banach algebra l1(S) is approximately amenable. It is also proved that if S is a Brandt semigroup over a group G with an index set I, then l1(S) is approximately amenable if and only if G is amenable. Moreover l1(S) is amenable if and only if G is amenable and I is finite. For a left cancellative foundation semigroup S with an identity such that for every Ma(S)-measurable subset B of S and s ∈ S the set sB is Ma(S)-measurable, it is proved that if the measure algebra Ma(S) is approximately amenable, then S is left amenable. Concrete examples are given to show that the converse is negative.

Multipliers on L(S), L(S)**, and LUC(S)* for a locally compact topological semigroup

We study compact and weakly compact multipliers on L(S), L(S)**, and LUC(S)*, where the latter is the dual of LUC(S). We show that a left cancellative semigroup S is left amenable if and only if there is a nonzero compact (or weakly compact) multiplier on L(S)**. We also prove that S is left amenable if and only if there is a nonzero compact (or weakly compact) multiplier on LUC(S)*.

Right simple subsemigroups and right subgroups of compact convergence semigroups

Clifford and Preston (1961) showed several important characterizations of right groups. It was shown in Roy and So (1998) that, among topological semigroups, compact right simple or left cancellative semigroups are in fact right groups, and the closure of a right simple subsemigroup of a compact semigroup is always a right subgroup. In this paper, it is shown that such results can be generalized in convergence semigroups. In the discussion of maximal right simple subsemigroups and maximal right subgroups of semigroups, generalization of the results that no two maximal right simple subsemigroups and maximal right subgroups of a convergence semigroup intersect, is also established.

An example of a regular inbreeding system with cubic ancestral growth that preserves some genetic variability

A specific regular inbreeding system of quadruple half-second cousin mating is considered. A regular inbreeding system can be thought of as a graph which satisfies certain natural homogeneity properties. Random walks Xn and Yn are introduced on the nodes of the graph; the event {Xn=Yn} is a renewal event by the homogeneity properties. In Arzberger (1985) it is shown that 1) graphs associated with left cancellative semigroups are regular, and 2) for regular systems, the population becomes genetically uniform if and only if the event {Xn=Yn} is recurrent. In Arzberger (1986) the system of quadruple half-second cousin mating is associated with a cancellative semigroup, thus the system is regular. In this paper we show that 1) An is asymptotically of the form cn3, where An is the number of ancestors n generations into the past, 2) {Xn=Yn} is not recurrent (this is shown by associating (Xn, Yn) with a random walk in Z3, 3) P[X3n=Y3n] is asymptotically of the form cn−3/2. Thus, in this example, genetic heterogeneity is maintained, with a cubic rate of growth for An, not by an exponential growth rate, as in all previous examples of regular inbreeding systems in which genetic heterogeneity is maintained.

On left cancellative semigroups

In this note we shall describe the structure of left cancellative semigroups. We shall also investigate the existence of idempotents and show that the existence of idempotents is related to the existence of maximal proper right ideals.