Topic
Cancellative semigroup
About: Cancellative semigroup is a research topic. Over the lifetime, 1320 publications have been published within this topic receiving 13319 citations.
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40 citations
TL;DR: In this paper, the authors investigate topologies that turn a directed graph E into a topological semigroup, where elements roughly correspond to possible paths in E. They show that in any such topology that is Hausdorff, G (E ) ∖ { 0 } must be discrete for any directed graph.
40 citations
TL;DR: In this paper, the rank of a finite semigroup is defined as the smallest number of elements required to generate the semigroup, and the problem of determining the maximum rank of the subsemigroup of the full transformation monoid (and of the symmetric inverse semigroup) is considered.
Abstract: The rank of a finite semigroup is the smallest number of elements required to generate the semigroup. A formula is given for the rank of an arbitrary (not necessarily regular) Rees matrix semigroup over a group. The formula is expressed in terms of the dimensions of the structure matrix, and the relative rank of a certain subset of the structure group obtained from subgroups generated by entries in the structure matrix, which is assumed to be in Graham normal form. This formula is then applied to answer questions about minimal generating sets of certain natural families of transformation semigroups. In particular, the problem of determining the maximum rank of a subsemigroup of the full transformation monoid (and of the symmetric inverse semigroup) is considered.
40 citations
TL;DR: In this article, it was shown that the number of defining equations of the tangent cone of a numerical semigroup ring is bounded by a value depending only on the width of the semigroup.
40 citations
TL;DR: In this paper, the structure of the semigroup tree and the regularities on the number of descendants of each node observed earlier by the first author are investigated and the question of what kind of chains appear in the tree and characterize the properties (like being (in)finite) thereof.
Abstract: In this paper we elaborate on the structure of the semigroup tree and the regularities on the number of descendants of each node observed earlier by the first author. These regularities admit two different types of behavior and in this work we investigate which of the two types takes place for some well-known classes of semigroups. Also we study the question of what kind of chains appear in the tree and characterize the properties (like being (in)finite) thereof. We conclude with some thoughts that show how this study of the semigroup tree may help in solving the conjecture of Fibonacci-like behavior of the number of semigroups with given genus.
40 citations