1,331 research outputs found

    Permutations of a semigroup that map to inverses

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    We investigate the question as to when the members of a finite regular semigroup may be permuted in such a way that each member is mapped to one of its inverses. In general this is not possible. However we reformulate the problem in terms of a related graph and, using an application of Hall’s Marriage Lemma, we show in particular that the finite full transformation semigroup does enjoy this property

    Finite regular semigroups with permutations that map elements to inverses

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    We give an account on what is known on the subject of permutation matchings, which are bijections of a finite regular semigroup that map each element to one of its inverses. This includes partial solutions to some open questions, including a related novel combinatorial problem

    The Biker-hiker problem

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    There are n travellers who have k bicycles and they wish to complete a journey in the shortest possible time. We investigate optimal solutions of this problem, showing they are characterized by a set of words in the Dyck language. Particular solutions with additional desirable properties are introduced and analysed

    Involution matchings, the semigroup of orientation-preserving and orientation-reversing mappings, and inverse covers of the full transformation semigroup

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    We continue the study of permutations of a fi nite regular semigroup that map each element to one of its inverses, providing a complete description in the case of semigroups whose idempotent generated subsemigroup is a union of groups. We show, in two ways, how to construct an involution matching on the semigroup of all transformations which either preserve or reverse orientation of a cycle. Finally, as an application, we use involution matchings to prove that when the base set has at least four members, a fi nite full transformation semigroup has no cover by inverse subsemigroups that is closed under intersection

    Production optimization by agents of differing work rates

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    We devise a scheme for producing, in the least possible time, nn identical objects with pp agents that work at differing speeds. This involves halting the process in order to transfer production across agent types. For the case of two types of agent, we construct a scheme based on the Euclidean algorithm that seeks to minimise the number of pauses in production.Comment: 19 pages, 2 figure

    Equationally defined classes of semigroups

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    We apply, in the context of semigroups, the main theorem from~\cite{higjac} that an elementary class C\mathcal{C} of algebras which is closed under the taking of direct products and homomorphic images is defined by systems of equations. We prove a dual to the Birkhoff theorem in that if the class is also closed under the taking of containing semigroups, some basis of equations of C\mathcal{C} is free of the \forall quantifier. Examples are given of EHP-classes that require more than two quantifiers in some equation of any equational basis.Comment: To appear in Semigroup Foru
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