Topic
Bicyclic semigroup
About: Bicyclic semigroup is a research topic. Over the lifetime, 1507 publications have been published within this topic receiving 19311 citations.
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TL;DR: In this paper, the authors studied the semigroup of partial co-final monotone bijective transformations of the set of positive integers and showed that it has algebraic properties similar to the bicyclic semigroup.
Abstract: In this paper we study the semigroup $\mathscr{I}_{\infty}^{
earrow}(\mathbb{N})$ of partial cofinal monotone bijective transformations of the set of positive integers $\mathbb{N}$. We show that the semigroup $\mathscr{I}_{\infty}^{
earrow}(\mathbb{N})$ has algebraic properties similar to the bicyclic semigroup: it is bisimple and all of its non-trivial group homomorphisms are either isomorphisms or group homomorphisms. We also prove that every locally compact topology $\tau$ on $\mathscr{I}_{\infty}^{
earrow}(\mathbb{N})$ such that $(\mathscr{I}_{\infty}^{
earrow}(\mathbb{N}),\tau)$ is a topological inverse semigroup, is discrete. Finally, we describe the closure of $(\mathscr{I}_{\infty}^{
earrow}(\mathbb{N}),\tau)$ in a topological semigroup.
24 citations
TL;DR: In this article, the algebraic structure of the semigroup of all 2x2 tropical matrices under multiplication is studied and a complete description of Green's relations and the idempotents and maximal subgroups of this semigroup is given.
24 citations
TL;DR: In this paper, it was shown that each inverse semigroup S can be embedded in some coset semigroup K(G) of a group G, modulo all subgroups of G, under the multiplication X*Y=smallest coset that constains XY.
Abstract: The setK(G) of all cosets X of a group G, modulo all subgroups of G, forms an inverse semigroup under the multiplication X*Y=smallest coset that constains XY. In this note we show that each inverse semigroup S can be embedded in some coset semigroupK(G). This follows from a result which shows that symmetric inverse semigroups can be embedded in the coset semigroups of suitable symmetric groups. We also give necessary and sufficient conditions on an inverse semigroup S in order that it should be isomorphic to someK(G).
24 citations
TL;DR: A monoid presentation of the dual symmetric inverse monoid is given and criteria for a monoid to be inverse when it is generated by completely regular elements are established.
Abstract: The dual symmetric inverse monoid is the inverse monoid of all isomorphisms between quotients of an n-set. We give a monoid presentation of and, along the way, establish criteria for a monoid to be inverse when it is generated by completely regular elements.
24 citations
TL;DR: In this article, the authors studied numerical semigroups of the form (x+S) + S with a numerical semigroup and a numerical setminus, and they studied the cases when the semigroup is symmetric, pseudo-symmetric, Arf and saturated.
Abstract: We study numerical semigroups of the form $(x+S)\cup\{0\}$ with $S$ a numerical semigroup and $x\in S\setminus\{0\}$. We pay special attention to the cases when $S$ is symmetric, pseudo-symmetric, Arf and saturated.
24 citations