Generalized transformation semigroups whose bi-ideals and quasi-ideals coincide.
A subsemigroup \(J\) of a semigroup \(S\) is a `quasi-ideal' of \(S\) if \(SJ\cap JS\subseteq J\) and it is a `bi-ideal' of \(S\) if \(JSJ\subseteq J\). Let \(X\) and \(Y\) be nonempty sets and let \(P(X,Y)\), \(T(X,Y)\), \(I(X,Y)\), \(M(X,Y)\), and \(E(X,Y)\) denote, respectively, the collection of all partial transformations of \(X\) into \(Y\), the collection of all transformations of \(X\) into \(Y\), the collection of all 1-1 partial transformations of \(X\) into \(Y\), the collection of all 1-1 transformations of \(X\) into \(Y\), and the collection of all surjections of \(X\) onto \(Y\). If \(S(X,Y)\) is any of the previous collections and \(\theta\in S(Y,X)\) then \((S(X,Y),\theta)\) denotes the semigroup where the product \(\alpha*\beta\) of any two elements \(\alpha,\beta\in(S(X,Y),\theta)\) is given by \(\alpha*\beta=\alpha\theta\beta\). Denote by \(\mathbf{BQ}\) the class of all semigroups whose collections of all bi-ideals and quasi-ideals coincide. The authors prove that \((S(X,Y),\theta)\in\mathbf{BQ}\) if \(S(X,Y)\) is any one of \(P(X,Y)\), \(T(X,Y)\), or \(I(X,Y)\). They go on to prove that if \(S(X,Y)\) is either \(M(X,Y)\) or \(E(X,Y)\), then \((S(X,Y),\beta)\in\mathbf{BQ}\) if and only if \(|X|=|Y|<\infty\).
scientific article; zbMATH DE number 2101755