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.     

Universal groups on semilattices of reversible cancellative semigroups

LetS be a semigroup which is a semilattice Ω of reversible cancellative semigroupsS _α, α∈Ω. This paper studies the relationship between the universal groupG onS and the universal groupsG _α onS _α. We also show that the universal homorphismsf _α∶S _α→G _α, α∈Ω fromS _α to the category of groups combine to a homomorphismf∶S→G ofS into the category of groups.     

Commutativity theorems for cancellative semigroups

Psomopoulos has proved that \([x^n, y] = [x, y^{n+1}]\) for a positive integer n implies commutativity in groups. Here we show that cancellative semigroups admitting commutators and satisfying the identity \([x^n, y] = [x, y^{n+k}]\) implies that the element \(y^k\) is central. The special case of \(k=1\) yields the above mentioned commutativity theorem. To accommodate negative exponents, we consider the functional equation \([f(x), y] = [x, g(y)f(y)] \) where f and g are unary functions satisfying certain formal syntactic rules and prove that in cancellative semigroups admitting commutators, the functional equation \([f(x), y] = [x, g(y)f(y)]\) implies that the element g(y) is central i.e. \(xg(y) = g(y)x\) for all x and y. By the way, these results are new even in group theory.     

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 a construction akin to the one devised by T. Tamura for N-semigroups.     

On identities of finite involution semigroups

We investigate the question of whether a finite involution semigroup is inherently nonfinitely based (INFB), which means that it is not contained in any finitely based locally finite variety. Although we fall short of a full characterization, we nevertheless clarify a number of interesting subcases.     

More on cancellative semigroups on manifolds

Communicated by D.R.Brown     

Prime and Semiprime Semigroup algebras of cancellative semigroups

Necessary and sufficient conditions are given for a semigroup algebra of a cancellative semigroup to be prime and semiprime. These conditions were proved necessary by Okninski; our contribution is to show that they are also sufficient. The techniques used in the proof are a new variation on the -methods which were developed originally for group algebras.

     

Radicals of algebras graded by cancellative linear semigroups

We consider algebras over a field of characteristic zero, and prove that the Jacobson radical is homogeneous in every algebra graded by a linear cancellative semigroup. It follows that the semigroup algebra of every linear cancellative semigroup is semisimple.

     

A cancellative amenable ascending union of nonamenable semigroups

We construct an example of a cancellative amenable semigroup which is the ascending union of semigroups, none of which are amenable.     

On commutation semigroups of dihedral groups

For any group G with g∈G, the right and left commutation semigroups associated with g are the mappings ρ(g) and λ(g) from G to G defined as (x)ρ(g)=[x,g] and (x)λ(g)=[g,x]. The set M(G) of all mappings from G to G forms a semigroup under composition of mappings. The right and left commutation semigroups of G, denoted P(G) and Λ(G), are the subsemigroups of M(G) generated by {ρ(g):g∈G} and {λ(g):g∈G}, respectively. In this paper, we develop explicit formulas for the orders of P(G) and Λ(G) when G=D _m , the dihedral group of order 2m. We apply these formulas to address the problems of determining when |P(G)|=|Λ(G)| and P(G)?Λ(G) and to derive proofs of several previous results of James Countryman (Ph.D. Thesis, University of Notre Dame, 1970) on commutation semigroups of pq groups.     

On equations in free semigroups and groups

The unsolvability of some algorithmic problems is proved for equations in free groups and semigroups, namely, some simple properties of the solutions of the equations are determined and the absence of an algorithm permitting the determination of whether an arbitrary equation in a free group or semigroup has a solution with the properties introduced is proved.Translated from Matematicheskie Zametki, Vol. 16, No. 5, pp. 717–724, November, 1974.     

On permutation properties in groups and semigroups

A semigroupS satisfiesPPn, thepermutation property of degree n (n≥2) if every product ofn elements inS remains invariant under some nontrivial permutation of its factors. It is shown that a semigroup satisfiesPP3 if and only if it contains at most one nontrivial commutator. Further a regular semigroup is a semilattice ofPP3 right or left groups, and a subdirect product ofPP3 semigroups of a simple type. A negative answer to a question posed by Restivo and Reutenauer is provided by a suitablePP3 group.