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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: It is proved that the set of all fuzzy congruences on a regular semigroup contained in δ H forms a modular lattice, where δH is the characteristic function of H and H is the H -equivalent relation on the semigroup.
23 citations
TL;DR: The semigroup of all operators T such that ( Tx, Tx )⩾( x, x ), for all elements of x of a finite-dimensional complex vector space with (, ) a given, possibly indefinite Hermitian form on that space, is the object under study in this article.
23 citations
TL;DR: In this paper, the question of whether the inverse of a generator of a bounded semigroup also generates a bounded semiigroup was studied on the Banach space and it was shown that the question must be answered negatively.
Abstract: In this paper we study the question whether $A^{-1}$ is the infinitesimal generator of a bounded $C_0$-semigroup if $A$ generates a bounded $C_0$-semigroup. If the semigroup generated by $A$ is analytic and sectorially bounded, then the same holds for the semigroup generated by $A^{-1}$. However, we construct a contraction semigroup with growth bound minus infinity for which $A^{-1}$ does not generate a bounded semigroup. Using this example we construct an infinitesimal generator of a bounded semigroup for which its inverse does not generate a semigroup. Hence we show that the question posed by deLaubenfels in 1988 must be answered negatively. All these examples are on Banach spaces. On a Hilbert space the question whether the inverse of a generator of a bounded semigroup also generates a bounded semigroup still remains open.
23 citations
TL;DR: McAlister's Theorem A and B as discussed by the authors states that every E-unitary inverse semigroup is isomorphic to a quotient of a P-semigroup by an idempotent-separating congruence.
Abstract: By an E-unitary inverse semigroup we mean an inverse semigroup in which the semilattice is a unitary subset. Such semigroups, elsewhere called ‘proper’ or ‘reduced’ inverse semigroups, have been the object of much recent study. Free inverse semigroups form a subclass of particular interest.An important structure theorem for E-unitary inverse semigroups has been obtained by McAlister [4, 5]. From a triple (G, ) consisting of a group G, a partially ordered set and a subset , satisfying certain conditions, he constructs an E-unitary inverse semigroup P(G, ). A semigroup of this type is called a P-semigroup. The structure theorem states that every E-unitary inverse semigroup is, to within isomorphism, of this form. A second theorem asserts that every inverse semigroup is isomorphic to a quotient of a Psemigroup by an idempotent-separating congruence. We refer below to these results as McAlister's Theorems A and B respectively. A triple (C, ) of the type used to construct a P-semigroup is here termed a “McAlister triple”. It is shown further, in [5], that there is essentially only one such triple corresponding to a given E-unitary inverse semigroup.
23 citations
TL;DR: In this article, a semigroup variety is called a variety of degree ≤ 2 if all its nilsemigroups are semigroups with zero multiplication and if all semigroup varieties of degree > 2 have zero multiplication unless they are upper-modular elements of the lattice.
Abstract: A semigroup variety is called a variety of degree ≤2 if all its nilsemigroups are semigroups with zero multiplication, and a variety of degree >2 otherwise. We completely determine all semigroup varieties of degree >2 that are upper-modular elements of the lattice of all semigroup varieties and find quite a strong necessary condition for semigroup varieties of degree ≤2 to have the same property.
23 citations