How can representation theories of inverse semigroups and lattices be united?
Let \(X\) be a set and \(\text{Rel}(X)\) the set of all binary relations on \(X\). For \(R\in\text{Rel}(X)\), let us put \(\text{Dom} (R)=\{x\mid \exists y:(x,y)\in R\}\) and define three operations \(\circ\), \({}^{-1}\) and \({\mathcal P}\) on \(X\) by: \(R\circ H= \{(x,y)\mid \exists z:(x,z)\in R\), \((z,y)\in H\}\), \(R^{-1}= \{(x,y)\mid (y,x)\in R\}\) and \({\mathcal P}(R)= \{(x,x)\mid x\in\text{Dom} (R)\}\). A binary relation \(R\) is called difunctional if \(R\circ R^{-1}\circ R=R\). This notion was introduced by \textit{J. Riguet} [Bull. Soc. Math. France 76, 114-154 (1948; Zbl 0033.00603)]. Let us denote by \(D(X)\) the set of all difunctional relations on \(X\). For \(R,H\in D(X)\) let us put \(R\bullet H=\bigcap \{Q\in D(X)\mid R\circ H\subseteq Q\}\). A partially ordered algebra \(\langle A;\cdot, {}^{-1},{}^*, \leq\rangle\) of type \((2,1,1)\) is called a regular partially ordered unary involuted semigroup if \(\langle A;\cdot, {}^{-1}\rangle\) is an involuted semigroup (that is \((xy)z=x(yz)\), \((x^{-1})^{-1}=x\) and \((xy)^{-1}= y^{-1}x^{-1})\), \(\leq\) is a partial order, the operations \(\cdot\), \({}^{-1}\) and \({}^*\) are monotonic, and the following identities hold: \((x^*)^*= x^*\), \(x^*x=x\), \(x^*y^*= y^*x^*\), \((x^*y^*)^*= x^*y^*\), \((xy)^*= (xy^*)^*\), \(x^*y\leq y\) and \(x^*\leq x^{-1}\). The main result of the article is the following Theorem 1. For any regular partially ordered unary involuted semigroup \(\langle A;\cdot, {}^{-1}, {}^*,\leq\rangle\), there exists an isomorphism \(\varphi\) of \(\langle A;\cdot, {}^{-1}, {}^*,\leq\rangle\) into \(\langle D(X);\bullet, {}^{-1},{\mathcal P},\subseteq\rangle\) for some set \(X\) such that, for any \(C\subseteq A\), \(\varphi(\inf C)=\bigcap_{a\in C}\varphi(a)\) whenever \(\inf C\) exists. This result generalizes simultaneously two well-known representation theorems: the Wagner-Preston theorem (about representation of inverse semigroups by inverse semigroups of partial one-to-one transformations) and the Whitman theorem (about representation of lattices by lattices of equivalence relations).
scientific article; zbMATH DE number 936966