Bicyclic semigroup

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

Möbius monoids and their connection to inverse monoids

Abstract: Mobius monoids are Mobius categories in the sense of Leroux having a single object. The paper explores several properties of Mobius monoids and inverse monoids as well as the links between them. The right cancellative, left rigid Mobius monoids have a special place in our study. The paper has two goals: first, an examination of Mobius functions and Mobius inversion under suitable conditions, and second, a development of special inverse monoids and inverse submonoids which arise from Mobius monoids and Mobius categories. The prototype of this development is the polycyclic monoid, more precisely the free monoid as a Mobius monoid. This leads us to the generalization of some significant results on polycyclic monoids as Meakin–Sapir’s result involving self-conjugate inverse submonoids, or as Jones-Lawson’s results involving gauge inverse submonoids of the polycyclic monoids. Regarding Mobius functions we create links between several types of these functions.

The Maximal Semilattice Decomposition of a Semigroup, Radicals and Nilpotency

Abstract: In paper [7] M. Petrich dealt with the maximal semilattice decomposition of a semigroup and he studied the classes of this decomposition. In the present paper a description of these classes and their products is given in terms of Luh Jiang completely prime radicals and faces of a semigroup S. Also the case of the commutative semigroup is discussed. The last, 5th section is self-containe d. Here a characterizati on of the class of all periodic semigroups with period 1, a characterization of the class of all periodic semigroups with index 1 and some characterizations of the class of all bands are given. We accomplished this using the mappings M-> Nt(M) (i = = 1,2, 3), where N±(M) (N2(M)) [N3(M)] is the set of all strongly (weakly) [almost] nilpotent elements with respect to the subset M of the semigroup S (see [9]).

On a semitopological extended bicyclic semigroup with adjoined zero

Abstract: In the paper it is shown that every Hausdorff locally compact semigroup topology on the extended bicyclic semigroup with adjoined zero $\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ is discrete, but on $\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ there exist $\mathfrak{c}$ many different Hausdorff locally compact shift-continuous topologies. Also, it is constructed on $\mathscr{C}_{\mathbb{Z}}^{\mathbf{0}}$ the unique minimal shift continuous topology and the unique minimal inverse semigroup topology.

On the structure of dimonoids

Abstract: J. -L. Loday introduced the notion of a dimonoid and constructed the free dimonoid. Cayley’s theorem for dimonoids states that every dimonoid is isomorphic to some transformation dimonoid. In this paper we propose another approach to constructing dimonoids which is based on using a semigroup operation. Several dimonoid-theoretical constructions are suggested, and it is shown that any dimonoid is isomorphically embedded into some dimonoid constructed from a semigroup. A similar result is obtained for dirings.

Ellis enveloping semigroup for almost canonical model sets

Abstract: This work deals with certain point patterns of an Euclidean space, for which the calculation of the Ellis enveloping semigroup of their associated dynamical systems is performed. The algebraic structure and the topology of the Ellis semigroup, as well as its action on the underlying space, are explicitly described. The present work is illustrated with the treatment of the vertex pattern of the so-called Amman-Beenker tiling of the plane.