Topic
Bicyclic semigroup
About: Bicyclic semigroup is a research topic. Over the lifetime, 1507 publications have been published within this topic receiving 19311 citations.
Papers published on a yearly basis
Papers
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TL;DR: In this article, a characterization of arithmetical invariants by the monoid of relations is presented, which is then extended to the monotone catenary degree and then applied to the explicit computation of invariants of semigroup rings.
Abstract: The investigation and classification of non-unique factorization phenomena has attracted some interest in recent literature. For finitely generated monoids, S.T. Chapman and P.A. Garc\'ia-S\'anchez, together with several co-authors, derived a method to calculate the catenary and tame degree from the monoid of relations. Then, in [1], the algebraic structure of this approach was investigated and the restriction to finitely generated monoids was removed. We now extend these ideas further to the monotone catenary degree and then apply all these results to the explicit computation of arithmetical invariants of semigroup rings.
[1] A. Philipp. A characterization of arithmetical invariants by the monoid of relations. Semigroup Forum, 81:424-434, 2010.
5 citations
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TL;DR: It is proved that for any positive integer n there exists an inverse semigroup ϒn of deficiency 1 and rank n + 1 such that ϒ n has exponential growth and it does not contain nonmonogenic free inverse subsemigroups.
Abstract: For any positive integer n > 1 we construct an example of inverse semigroup with n generators and n - 1 defining relations which has cubic growth and at least n generators in any presentation. This semigroup has the same set of identities as the free monogenic inverse semigroup. In particular, we give the first example of a one relation nonmonogenic inverse semigroup having polynomial growth. We also prove that for any positive integer n there exists an inverse semigroup ϒn of deficiency 1 and rank n + 1 such that ϒn has exponential growth and it does not contain nonmonogenic free inverse subsemigroups. Furthermore, ϒn satisfies the identity [[x, y], [z, t]]2 = [[x, y], [z, t]] of quasi-solvability and it contains a free subsemigroup of rank 2.
5 citations
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TL;DR: In this paper, it was shown that no algorithm can determine from an arbitrary recursive system of semigroup identities whether the variety defined by this system is finitely based, i.e., it is not finitely-based.
Abstract: No algorithm determines from an arbitrary recursive system of semigroup identities whether the variety defined by this system is finitely based.
5 citations
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TL;DR: In this paper, it was shown that a tightly connected fundamental inverse semigroup with no isolated nontrivial subgroups is lattice determined ''modulo semilattices'' and if the inverse monoid of a semigroup whose partial automorphism monoid is isomorphic to that of the semigroup is a monoid consisting of all isomorphisms between its inverse subsemigroups, then either the two monoids are isomorphic or they are dually isomorphic chains relative to the natural order.
Abstract: The partial automorphism monoid of an inverse semigroup is an inverse monoid consisting of all isomorphisms between its inverse subsemigroups. We prove that a tightly connected fundamental inverse semigroup $S$ with no isolated nontrivial subgroups is lattice determined `modulo semilattices' and if $T$ is an inverse semigroup whose partial automorphism monoid is isomorphic to that of $S$, then either $S$ and $T$ are isomorphic or they are dually isomorphic chains relative to the natural partial order; a similar result holds if $T$ is any semigroup and the inverse monoids consisting of all isomorphisms between subsemigroups of $S$ and $T$, respectively, are isomorphic. Moreover, for these results to hold, the conditions that $S$ be tightly connected and have no isolated nontrivial subgroups are essential.
5 citations
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TL;DR: In this paper, it was shown that every complete algebraic lattice can be the lattice of filters of a semigroup and that every semigroup is a homomorphic image of a finite semigroup whose lattice is boolean and which belongs to the pseudovariety generated by the original semigroup.
Abstract: A filter in a semigroup is a subsemigroup whose complement is an ideal. (Alternatively, in a quasiordered semigroup, a slightly more general definition can be given.) We prove a number of results related to filters in a semigroup and the lattice of filters of a semigroup. For instance, we prove that every complete algebraic lattice can be the lattice of filters of a semigroup. We prove that every finite semigroup is a homomorphic image of a finite semigroup whose lattice of filters is boolean and which belongs to the pseudovariety generated by the original semigroup. We describe filter lattices of some well-known semigroups such as full transformation semigroups of finite sets (which are three-element chains) and free semigroups (which are boolean).
5 citations