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Semigroups of transformations preserving an equivalence relation and a cross-section. (English) Zbl 1068.20061

Summary: For a set \(X\), an equivalence relation \(\rho\) on \(X\), and a cross-section \(R\) of the partition \(X/\rho\) induced by \(\rho\), consider the semigroup \(T(X,\rho,R)\) consisting of all mappings \(a\) from \(X\) to \(X\) such that \(a\) preserves both \(\rho\) (if \((x,y)\in\rho\) then \((xa,ya)\in\rho\)) and \(R\) (if \(r\in R\) then \(ra\in R\)). The semigroup \(T(X,\rho,R)\) is the centralizer of the idempotent transformation with kernel \(\rho\) and image \(R\). We determine the structure of \(T(X,\rho,R)\) in terms of Green’s relations, describe the regular elements of \(T(X,\rho,R)\), and determine the following classes of the semigroups \(T(X,\rho,R)\): regular, abundant, inverse, and completely regular.

MSC:

20M20 Semigroups of transformations, relations, partitions, etc.
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