Recent developments in the theory of skew lattices
A skew lattice is an algebra \(\langle A; \wedge, \vee\rangle\) where \(\wedge\) and \(\vee\) are associative binary operations (not necessarily commutative) satisfying the absorption identities: \[ \begin{aligned} x \wedge (x\vee y) &= x= (y \vee x)\wedge x,\\ x\vee (x\wedge y) &= x= (y\wedge x)\vee x. \end{aligned} \] Every lattice is a skew lattice. Every rectangular band can be made into a skew lattice by setting \(x\wedge y= xy\) and \(x\vee y= yx\). The author provides a systematic review of the work on skew lattices starting with the work of P. Jordan in the fifties up to his own interesting contributions in the nineties. The paper is written for a semigroup literate audience. Nonassociative generalizations of lattices (which are not semigroup theoretic in nature, e.g., weakly nonassociative lattices) are not discussed. There appear to be some original results hidden in the excellent exposition; for instance, the free skew lattice on two generators is infinite.
scientific article; zbMATH DE number 836804
scientific article