Cancellative Semigroups Research Articles

Connes-amenability of bidual and weighted semigroup algebras

We investigate the notion of Connes-amenability, introduced by Runde in [10], for bidual algebras and weighted semigroup algebras. We provide some simplifications to the notion of a $\sigma WC$-virtual diagonal, as introduced in [13], especially in the case of the bidual of an Arens regular Banach algebra. We apply these results to discrete, weighted, weakly cancellative semigroup algebras, showing that these behave in the same way as ${ C}^*$-algebras with regards Connes-amenability of the bidual algebra. We also show that for each one of these cancellative semigroup algebras $l^1(S,\omega)$, we have that $l^1(S,\omega)$ is Connes-amenable (with respect to the canonical predual $c_0(S)$) if and only if $l^1(S,\omega)$ is amenable, which is in turn equivalent to $S$ being an amenable group, and the weight satisfying a certain restrictive condition. This latter point was first shown by Grønnbæk in [6], but we provide a unified proof. Finally, we consider the homological notion of injectivity, and show that here, weighted semigroup algebras do not behave like ${ C}^*$-algebras.

Lattices embeddable in subsemigroup lattices. II. Cancellative semigroups

Repnitskii proved that any lattice embeds in a subsemigroup lattice of some commutative, cancellative, idempotent free semigroup with unique roots. In that proof, use is made of a result by Bredikhin and Schein stating that any lattice embeds in a suborder lattice of suitable partial order. Here, we present a direct proof of Repnitskii’s result which is independent of Bredikhin-Schein’s, thus giving the answer to the question posed by Shevrin and Ovsyannikov.

Semigroup gradings on associative rings

The main aim of this paper is the adaptation of the results of the papers [Y.A. Bahturin, S.K. Sehgal, M.V. Zaicev, Group gradings on associative algebras, J. Algebra 241 (2) (2001) 677–698; Y.A. Bahturin, S.K. Sehgal, M.V. Zaicev, Finite-dimensional graded simple algebras, preprint, and M.V. Zaĭtsev, S.K. Segal, Finite gradings of simple Artinian rings, Vestnik Moskov. Univ. Ser. I Mat. Mekh., vol. 3, 2001, pp. 21–24, 77 (in Russian); translation in Moscow Univ. Math. Bull. 56 (3) (2001) 21–24] about the structure of graded simple rings from the gradings by groups to the gradings by cancellative semigroups.

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.

Levi’s Commutator Theorems for Cancellative Semigroups

A conjecture of Padmanabhan, on provability in cancellative semigroups, is addressed. Several of Levi’s group theory commutator theorems are proved for cancellative semigroups. The proofs, found by automated deduction, support the conjecture.

Levi’s Commutator Theorems for Cancellative Semigroups

For every semigroup of finite exponent whose chains of idempotents are uniformly bounded we construct an identity which holds on this semigroup but does not hold on the variety of all idempotent semigroups. This shows that the variety of all idempotent semigroups E is not contained in any finitely generated variety of semigroups. Since E is locally finite and each proper subvariety of E is finitely generated [1, 3, 4], the variety of all idempotent semigroups is a minimal example of an inherently non-finitely generated variety.

On identities in groups of fractions of cancellative semigroups

To solve two problems of Bergman stated in 1981, we construct a group G G such that G G contains a free noncyclic subgroup (hence, G G satisfies no group identity) and G G , as a group, is generated by its subsemigroup that satisfies a nontrivial semigroup identity.

The rank of a commutative cancellative semigroup

For a commutative cancellative semigroup S , we define the rank of S intrinsically. This definition implies that the rank of S equals the usual rank of its group of quotients. We also characterize the rank in terms of embeddability into a rational vector space of the greatest power cancellative image of S.

Prime Ideals of Cancellative Semigroups#

For a class of prime ideals P of a cancellative semigroup S it is shown that the factors S/P have a structure of a monomial semigroup over a group. Consequences for the semigroup algebras K[S] are discussed.

Commutative cancellative semigroups of finite rank

The rank of a commutative cancellative semigroup S is the cardinality of a maximal independent subset of S. Commutative cancellative semigroups of finite rank are subarchimedean and thus admit a Tamura-like representation. We characterize these semigroups in several ways and provide structure theorems in terms of ac onstruction akin to the one devised by T. Tamura forN-semigroups.

SEMIPRIME QUADRATIC ALGEBRAS OF GELFAND–KIRILLOV DIMENSION ONE

We consider algebras over a field K with a presentation K<x1,…,xn:R>, where R consists of [Formula: see text] square-free relations of the form xixj=xkxl with every monomial xixj, i≠j, appearing in one of the relations. The description of all four generated algebras of this type that satisfy a certain non-degeneracy condition is given. The structure of one of these algebras is described in detail. In particular, we prove that the Gelfand–Kirillov dimension is one while the algebra is noetherian PI and semiprime in case when the field K has characteristic zero. All minimal prime ideals of the algebra are described. It is also shown that the underlying monoid is a semilattice of cancellative semigroups and its structure is described. For any positive integer m, we construct non-degenerate algebras of the considered type on 4m generators that have Gelfand–Kirillov dimension one and are semiprime noetherian PI algebras.

Baer-Levi semigroups of partial transformations

Let X be an infinite set and suppose א0 ≤ q ≤ |X|. The Baer-Levi semigroup on X is the set of all injective ‘total’ transformations α: X → X such that |X\Xα| = q. It is known to be a right simple, right cancellative semigroup without idempotents, its automorphisms are “inner”, and some of its congruences are restrictions of Malcev congruences on I(X), the symmetric inverse semigroup on X. Here we consider algebraic properties of the semigroup consisting of all injective ‘partial’ transformations α of X such that |X\Xα| = q: in particular, we descried the ideals and Green's relations of it and some of its subsemigroups.

On Homogeneity of Radicals of Semigroup Algebras

We answer a known question showing that not all radicals of semigroup Q algebras of Q0 -complete commutative, cancellative and torsion-free semigroups are homogeneous.

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)*.

Sum-Free Subsets of Right Cancellative Semigroups

For a natural number k≥ 2 letρ=ρ(k) be the smallest natural number which does not dividek− 1. We show that for any subset A of a right cancellative semigroup S which contains no solutions of the equation x1+⋯ +xk=y there is an element s inS such that the setsA, A+s, . ,A + (ρ− 1)sare pairwise disjoint. In particular, if S is finite, such a set A has at most | S |/ρ elements. This estimate is sharp.

On equivalence of semigroup identities

For a given relation $\rho$ on a free semigroup ${\mathcal F}$ we describe the smallest cancellative fully invariant congruence ${\rho}^{\sharp}$ containing $\rho$. Two semigroup identities are s-equivalent if each of them is a consequence of the other on cancellative semigroups. If two semigroup identities are equivalent on groups, it is not known if they are s-equivalent. We give a positive answer to this question for all binary semigroup identities of the degree less or equal to 5. A poset of corresponding varieties of groups is given.

Some Semigroup Laws in Groups

AbstractA challenge by R. Padmanabhan to prove by group theory the commutativity of cancellative semigroups satisfying a particular law has led to the proof of more general semigroup laws being equivalent to quite simple ones.

An analogue for semigroups of a group problem of P. Erdös and B. H. Neumann

In response to a question posed by P. Erdös, B. H. Neumann showed that in a group with every subset of pairwise noncommuting elements finite there is a bound on the size of these sets. Recently, H. E. Bell, A. A. Klein and the first author showed that a similar result holds for rings. However in the case of semigroups, finiteness of subsets of pairwise noncommuting elements does not assure the existence of a bound for their size. The largest class of semigroups in which we found Neumann's result valid are cancellative semigroups.

Umbral Calculus and Cancellative Semigroup Algebras

We describe some connections between three different fields: combinatorics (umbral calculus), functional analysis (linear functionals and operators) and harmonic analysis (convolutions on group-like structures). Systematic usage of cancellative semigroups, their convolution algebras, and tokens between them provides a common language for description of objects from these three fields.

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.