2,552 research outputs found

    Cross-connections and variants of the full transformation semigroup

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    Cross-connection theory propounded by K. S. S. Nambooripad describes the ideal structure of a regular semigroup using the categories of principal left (right) ideals. A variant TXθ\mathscr{T}_X^\theta of the full transformation semigroup (TX,)(\mathscr{T}_X,\cdot) for an arbitrary θTX\theta \in \mathscr{T}_X is the semigroup TXθ=(TX,)\mathscr{T}_X^\theta= (\mathscr{T}_X,\ast) with the binary operation αβ=αθβ\alpha \ast \beta = \alpha\cdot\theta\cdot\beta where α,βTX\alpha, \beta \in \mathscr{T}_X. In this article, we describe the ideal structure of the regular part Reg(TXθ)Reg(\mathscr{T}_X^\theta) of the variant of the full transformation semigroup using cross-connections. We characterize the constituent categories of Reg(TXθ)Reg(\mathscr{T}_X^\theta) and describe how they are \emph{cross-connected} by a functor induced by the sandwich transformation θ\theta. This lead us to a structure theorem for the semigroup and give the representation of Reg(TXθ)Reg(\mathscr{T}_X^\theta) as a cross-connection semigroup. Using this, we give a description of the biordered set and the sandwich sets of the semigroup

    Cross-connections of the singular transformation semigroup

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    Cross-connection is a construction of regular semigroups using certain categories called normal categories which are abstractions of the partially ordered sets of principal left (right) ideals of a semigroup. We describe the cross-connections in the semigroup Sing(X)Sing(X) of all non-invertible transformations on a set XX. The categories involved are characterized as the powerset category P(X)\mathscr{P}(X) and the category of partitions Π(X)\Pi(X). We describe these categories and show how a permutation on XX gives rise to a cross-connection. Further we prove that every cross-connection between them is induced by a permutation and construct the regular semigroups that arise from the cross-connections. We show that each of the cross-connection semigroups arising this way is isomorphic to Sing(X)Sing(X). We also describe the right reductive subsemigroups of Sing(X)Sing(X) with the category of principal left ideals isomorphic to P(X)\mathscr{P}(X). This study sheds light into the more general theory of cross-connections and also provides an alternate way of studying the structure of Sing(X)Sing(X)

    The faunal remains from Shaqadud

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