Flat coset decompositions of skew lattices (Q283331)

Skew lattices are a non-commutative generalization of lattices. More precisely, an algebra \(L=(L;\vee,\wedge)\) of type \((2,2)\) is called a \textit{skew lattice}, if both operations \(\vee\) and \(\wedge\) are associative and satisfy the absorption identities \(x\wedge (x\vee y)=x=(y\vee x)\wedge x\) and their dual. Moreover, it is easy to show that both operations \(\vee\) and \(\wedge\) are idempotent. Therefore, the both reducts \(L_\vee =(L;\vee)\) and \(L_\wedge =(L;\wedge)\) of \(L\) are bands, i.e. semigroups consisting solely of idempotents. In the study of skew lattices, especially the following three Green equivalences are significant: \(\theta_d\), \(\theta_r\) and \(\theta_\ell\), where \(x\theta_d y\) iff \(x\wedge y\wedge x=x\) and \(y\wedge x\wedge y=y\), \(x\theta_r y\) iff \(x\wedge y=y\) and \(y\wedge x=x\) and, \(x\theta_\ell y\) iff \(x\wedge y=x\) and \(y\wedge x=y\). By \textit{J. Leech}'s theorems [Algebra Univers. 26, No. 1, 48--72 (1989; Zbl 0669.06006)], a skew lattice is a lattice of its Green \(\theta_d\)-classes which form its maximal rectangular subalgebras (i.e. satisfying \(x\wedge y=y\vee x\)). Moreover, \(L/\theta_d\) is the maximal lattice image of \(L\) The second theorem says that every skew lattice embeds into a direct product of a left-handed skew lattice and a right-handed one. (\(L\) is right-handed, if \(\theta_d=\theta_r.\) Left-handed algebras can be obtained similarly.) A skew lattice is called \textit{flat} if it is either left- or right-handed. The authors investigate the coset structure of flat skew lattices.