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 article, the semigroup and adjoint semigroup for a class of linear functional differential equations with infinite delays were considered, which includes certain linear Volterra integro-differential equations.
Abstract: We consider the semigroup and adjoint semigroup for a class of linear functional differential equations with infinite delays, which includes certain linear Volterra integro-differential equations. In particular, we show that by an appropriate choice of the state space the semigroup constructed by Miller [13] can be considered the adjoint semigroup of the semigroup constructed by Barbu and Grossman [2]. This provides a useful characterization of Miller’s semigroup which can be applied to obtain additional information about the semigroup defined by Barbu and Grossman.
33 citations
TL;DR: In this article, it was shown that the identities of the Kauffman monoid are nonfinitely based for any semigroup under either of its natural involutions, and this result holds also for the case when the semigroup is considered as an involution semigroup.
Abstract: We prove a sufficient condition under which a semigroup admits no finite identity basis. As an application, it is shown that the identities of the Kauffman monoid $${\mathcal{K}_n}$$
are nonfinitely based for each $${n \geq 3}$$
. This result holds also for the case when $${\mathcal{K}_n}$$
is considered as an involution semigroup under either of its natural involutions.
33 citations
TL;DR: In this article, the authors introduce algebraic dynamical systems, which consist of an action of a right LCM semigroup by injective endomorphisms of a group and describe it as a semigroup C*-algebra.
Abstract: We introduce algebraic dynamical systems, which consist of an action of a right LCM semigroup by injective endomorphisms of a group. To each algebraic dynamical system we associate a C*-algebra and describe it as a semigroup C*-algebra. As part of our analysis of these C*-algebras we prove results for right LCM semigroups. More precisely we discuss functoriality of the full semigroup C*-algebra and compute its K-theory for a large class of semigroups. We introduce the notion of a Nica-Toeplitz algebra of a product system over a right LCM semigroup, and show that it provides a useful alternative to study algebraic dynamical systems.
33 citations
TL;DR: The notion of an ordered 2-complex was introduced by Nambooripad and Pastijn as discussed by the authors, who showed that any ordered groupoid can be represented as the fundamental groupoid of a 2-composition.
Abstract: Work of Ehresmann and Schein shows that an inverse semigroup can be viewed as a groupoid with an order structure; this approach was generalized by Nambooripad to apply to arbitrary regular semigroups. This paper introduces the notion of an ordered 2-complex and shows how to represent any ordered groupoid as the fundamental groupoid of an ordered 2-complex. This approach then allows us to construct a standard 2-complex for an inverse semigroup presentation. Our primary applications are to calculating the maximal subgroups of an inverse semigroup which, under our topological approach, turn out to be the fundamental groups of the various connected components of the standard 2-complex. Our main results generalize results of Haatja, Margolis, and Meakin giving a graph of groups decomposition for the maximal subgroups of certain regular semigroup amalgams. We also generalize a theorem of Hall by showing the strong embeddability of certain regular semigroup amalgams as well as structural results of Nambooripad and Pastijn on such amalgams.
33 citations
TL;DR: In this paper, the regularity preserving elements of regular semigroups are defined, i.e., the elements a such that (S, ∈ S) is also regular.
Abstract: Let S be a regular semigroup, and let a ∈ S . Then a variant of S with respect to a is a semigroup with underlying set S and multiplication \circ defined by x \circ y = xay . In this paper, we characterise the regularity preserving elements of regular semigroups; these are the elements a such that (S,\circ) is also regular. Hickey showed that the set of regularity preserving elements can function as a replacement for the unit group when S does not have an identity. As an application, we characterise the regularity preserving elements in certain Rees matrix semigroups. We also establish connections with work of Loganathan and Chandrasekaran, and with McAlister's work on inverse transversals in locally inverse semigroups. We also investigate the structure of arbitrary variants of regular semigroups concentrating on how the local structure of a semigroup affects the structure of its variants.
32 citations