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Embedding theorems and maximal subsemigroups of some linear transformation semigroups with restricted range. (English) Zbl 1496.20116

Ukr. Math. J. 73, No. 12, 1985-1996 (2022) and Ukr. Mat. Zh. 73, No. 12, 1714-1722 (2021).
Summary: Let \(V\) be a vector space over a field and let \(T(V)\) denote the semigroup of all linear transformations from \(V\) into \(V\). For a fixed subspace \(W\) of \(V\), let \(F(V,W)\) be the subsemigroup of \(T(V\) ) formed by all linear transformations \(\alpha\) from \(V\) into \(W\) such that \(V \alpha \subseteq W \alpha \). We prove that any regular semigroup \(S\) can be embedded in \(F(V,W)\) with \(\dim (V) = |S^1|\) and \(\dim (W) = |S|,\) and determine all maximal subsemigroups of \(F(V,W)\) in the case where \(W\) is a finite-dimensional subspace of \(V\) over a finite field.

MSC:

20M20 Semigroups of transformations, relations, partitions, etc.
15A30 Algebraic systems of matrices
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References:

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