Regular semigroups with inverse transversals.
An inverse subsemigroup \(S^\circ\) of a regular semigroup \(S\) is called an inverse transversal of \(S\) if \(S^\circ\) contains an unique inverse \(x^\circ\) for every \(x\in S\). Let \(E(S)\) denote the set of idempotents of \(S\). A subsemigroup \(T\) of \(S\) is full if \(E(S)\subseteq T\). If \(C\) is the regular subsemigroup generated by \(E(S)\) then denote \(\prec\!e\!\succ=\langle E(eCe)\rangle\) for each \(e\in E(C)\) and define for each \(a\in S\) a partial mapping \(\lambda_a\colon\prec\!aa^\circ\!\succ\to\prec\! a^\circ a\!\succ\) by \(x\mapsto a^\circ xa\). Denote \(I=\{e\in C:e=ee^\circ\}\), \(\Lambda=\{f\in C:f=f^\circ f\}\) and for \(e\in I\), \(f\in\Lambda\) let \(T_{e,f}\) be the set of all isomorphisms \(\alpha\colon\prec\!e\!\succ\to\prec\!f\!\succ\) such that \((eE^\circ)\alpha=E^\circ f\). Denote \(T_{C,C^\circ}=\bigcup T_{e,f}\) and define for \(\alpha\in T_{e,f}\), \(\beta\in T_{g,h}\) the product \(\alpha\circ\beta=\alpha\lambda_f^{-1}\lambda_{fg}\lambda_g^{-1}\beta\). Let \(K\) be an inverse semigroup, \(C\) a semiband with inverse transversal \(C^\circ\) such that \(E(K)\subseteq C^\circ\) and \(T\) a full regular subsemigroup of \(T_{C,C^\circ}\). For any \(a\in K\) let \(\varphi_a\) be a transformation on \(I\cup\Lambda\) such that \(l\varphi_t=t^\circ lt\), \(\varphi_a\varphi_b=\varphi_{ab}\) for any \(l\in I\cup\Lambda\), \(t\in C^\circ\), \(a,b\in K\). Define \(T*K=\{(\alpha,a):\alpha\in T_{e,f}\), \(a\in K\), \(e^\circ=aa^{-1}\), \(f^\circ=a^{-1}a\), \((e^\circ le^\circ)\alpha^{\circ\circ}=l\varphi_a\}\) (it is required that all products exist for every \(l\in I\cup\Lambda\), \(c\alpha^{\circ\circ}=a^{-1}ca\), \(c\in\text{dom\,}\alpha^{\circ\circ}\cap C^\circ\)). The regular semigroup \(T*K\) has inverse transversal \(T^\circ*K\simeq K\) and \((T*K)/\mu_{T*K}\simeq T\); here \(\mu_{T*K} \) is the maximal idempotent-separating congruence on \(T*K\). If \(S\) is a regular semigroup with an inverse transversal \(S^\circ\) and \(C=\langle E(S)\rangle\), then \(S\simeq T*S^\circ\), where \(T=\{\lambda_s\), \(s\in S\}\) is a full regular subsemigroup of \(T_{C,C^\circ}\) with inverse transversal \(T^\circ=\{\lambda_s\), \(s\in S\}\subseteq T_{C,C^\circ}^\circ\).
scientific article; zbMATH DE number 5118912