Bicyclic semigroup

About: Bicyclic semigroup is a research topic. Over the lifetime, 1507 publications have been published within this topic receiving 19311 citations.

Hypercyclic abelian semigroup of matrices on Cn and Rn and k-transitivity (k ≥ 2)

Abstract: We prove that the minimal number of matrices on Cn required to forma hypercyclic abelian semigroup on Cn is n+1. We also prove that theaction of any abelian semigroup finitely generated by matrices on Cnor Rn is never k-transitive for k 2. These answer questions raised byFeldman and Javaheri.

Computing the ideal structure of finite semigroups

Abstract: In this paper algorithms are described for computing the equivalence classes induced by the Green relations on the elements of a finite semigroup.

Growth and existence of identities in a class of finitely presented inverse semigroups with zero

Abstract: We prove that a finitely presented Rees quotient of a free inverse semigroup has polynomial or exponential growth, and that the type of growth is algorithmically recognizable. We prove that such a semigroup has polynomial growth if and only if it satisfies a certain semigroup identity. However we give an example of such a semigroup which has exponential growth and satisfies some nontrivial identity in signature with involution.

The Semigroups B 2 and B 0 are Inherently nonfinitely Based, as restriction Semigroups.

Abstract: The five-element Brandt semigroup B2 and its four-element subsemigroup B0, obtained by omitting one nonidempotent, have played key roles in the study of varieties of semigroups Regarded in that fashion, they have long been known to be finitely based The semigroup B2 carries the natural structure of an inverse semigroup Regarded as such, in the signature {⋅, -1}, it is also finitely based It is perhaps surprising, then, that in the intermediate signature of restriction semigroups — essentially, "forgetting" the inverse operation x ↦ x-1 and retaining the induced operations x ↦ x+ = xx-1 and x ↦ x* = x-1x — it is not only nonfinitely based but inherently so (every locally finite variety that contains it is also nonfinitely based) The essence of the nonfinite behavior is actually exhibited in B0, which carries the natural structure of a restriction semigroup, inherited from B2 It is again inherently nonfinitely based, regarded in that fashion It follows that any finite restriction semigroup on which the two unary operations do not coincide is nonfinitely based Therefore for finite restriction semigroups, the existence of a finite basis is decidable "modulo monoids" These results are consequences of — and discovered as a result of — an analysis of varieties of "strict" restriction semigroups, namely those generated by Brandt semigroups and, more generally, of varieties of "completely r-semisimple" restriction semigroups: those semigroups in which no comparable projections are related under the generalized Green relation 𝔻 For example, explicit bases of identities are found for the varieties generated by B0 and B2

On deterministic finite automata and syntactic monoid size, continued

Abstract: We continue our investigation on the relationship between regular languages and syntactic monoid size. In this paper we confirm the conjecture on two generator transformation semigroups. We show that for every prime n ≥ 7 there exist natural numbers k and l with n = k + l such that the semigroup Uk, l is maximal w.r.t. its size among all (transformation) semigroups which can be generated with two generators. This significantly tightens the bound on the syntactic monoid size of languages accepted by n-state deterministic finite automata with binary input alphabet. As a by-product of our investigations we are able to determine the maximal size among all semigroups generated by two transformations, where one is a permutation with a single cycle and the other is a non-bijective mapping.