10 research outputs found

    Regular elements and Green's relations in Generalised Linear Transformation Semigroups

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    If V and W are vector spaces over the same field, we let P(V,W) denote the set of all partial linear transformations from V into W (that is, all linear mappings whose domain and range are subspaces of V and W, respectively). If θP(W,V)\theta\in P(W,V), then P(V,W) is a so-called `generalised semigroup' of linear transformations under the `sandwich operation': αβ=αθβ\alpha *\beta=\alpha\circ\theta\circ\beta, for each α,βP(V,W)\alpha,\beta\in P(V,W). We denote this semigroup by P(V,W,θ)P(V,W,\theta) and, in this paper, we characterise Green's relations on it: that is, we study equivalence relations which determine when principal left (or right, or 2-sided) ideals in P(V,W,θ)P(V,W,\theta) are equal. This is related to a problem raised by Magill and Subbiah in 1975. We also discuss the same idea for important subsemigroups of P(V,W,θ)P(V,W,\theta) and characterise when these semigroups satisfy certain regularity conditions.Fundação para a Ciência e a Tecnologia (FCT
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