The congruences on the semigroup of balanced transformations of an infinite set
\(T(X)\) denotes the full transformation semigroup on an infinite set \(X\). For \(\alpha\in T(X)\), let \(r(\alpha)=|X\alpha|\), \(d(\alpha)=|X\setminus X\alpha|\), \(s(\alpha)=|\{x\in X:x\alpha\neq x\}|\), and \(c(\alpha)=|\cup\{y\alpha^{-1}:|y\alpha^{-1}|\geq 2\}|\). Next, let \(H=\{\alpha\in T(X):d(\alpha)=s(\alpha)=c(\alpha)\geq\aleph_0\}\). \(H\) is a subsemigroup of \(T(X)\) and is referred to as the semigroup of balanced transformations on \(X\). In this paper, the authors describe all the congruences on the semigroup \(H\). Let \(k=|X|\) and let \(k'\) denote the immediate successor of \(k\). If \(\aleph_0\leq\delta\leq k\) and \(2\leq\eta\leq k'\), then \(H(\delta,\eta)=\{\alpha\in H:d(\alpha)\geq\delta\) and \(r(\alpha)<\eta\}\) is an ideal of \(H\) and every ideal of \(H\) has this form. The Rees congruence on \(H\) determined by this ideal is denoted by \(H(\delta,\eta)^*\). Let \(D(\alpha,\beta)=\{x\in X:x\alpha\neq x\beta\}\) and \(\text{dr}(\alpha,\beta)=\max\{|D(\alpha,\beta)\alpha|,|D(\alpha,\beta)\beta|\}\). Then let \(\Delta_\xi=\{(\alpha,\beta)\in T(X)\times T(X):\text{dr}(\alpha,\beta)<\xi\}\). The authors show, for example, that if \(\xi_i\), \(\eta_i\) are infinite cardinal numbers with the possible exception of \(\xi_r\) which may be 1 such that \(\xi_r<\cdots<\xi_1\leq\eta_1<\cdots<\eta_r\leq k\), then \[ H(k,\eta_1)^*\cup[\Delta_{\xi_1}\cap H(k,\eta_2)^*]\cup\cdots\cup[\Delta_{r-1}\cap H(k,\eta_r)^*]\cup[\Delta_{\xi_r}\cap(H\times H)] \] is a congruence on \(H\) and every congruence on \(H\) which satisfies certain conditions is of this form.
scientific article; zbMATH DE number 1565852
scientific article