Recent developments in the theory of skew lattices
type: Item
instance of: scholarly article
title: Recent developments in the theory of skew lattices
zbMATH Open document ID: 0844.06003
DOI: 10.1007/BF02574077
published in: Semigroup Forum
publication date: 1996-08-19T00:00:00Z
full work available at URL: https://eudml.org/doc/135429
review text: 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.
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 20M99
Mathematics Subject Classification ID: 06-02
zbMATH DE Number: 836804
zbMATH Keywords: survey
zbMATH Keywords: skew lattice
zbMATH Keywords: rectangular band
author: Jonathan Leech
MaRDI profile type: Publication
cites work: Skew Boolean algebras and discriminator varieties
cites work: Q3934450
cites work: Boolean skew algebras
cites work: Zur Charakterisierung distributiver Schiefverbände
cites work: Zerlegungshomomorphismen in Schrägverbänden
cites work: Q4126563
cites work: Halbgruppen von Idempotenten und nichtkommutative Verbände.
cites work: Q4769085
cites work: The structure of idempotent semigroups. I
cites work: Skew lattices in rings
cites work: Skew Boolean algebras
cites work: Normal skew lattices
cites work: The Geometric Structure of Skew Lattices
cites work: Zur Theorie der niochtkommutativen Verbände. I
cites work: Q3659940
cites work: Idempotent distributive semirings. II
cites work: Q4148155
cites work: Q3996704
cites work: Restrictive bisemigroups
cites work: Pseudosemilattices and pseudolattices
cites work: Q3699738
OpenAlex ID: W2013122876
Recommended article: On strongly symmetric skew lattices
Recommended article: The Geometric Structure of Skew Lattices
Recommended article: Skew lattices in rings
Recommended article: Skew lattices
Recommended article: On the coset structure of a skew lattice
Recommended article: Q2998596
Recommended article: On residuated skew lattices
Recommended article: Q3701482
Recommended article: On the coset laws for skew lattices
Recommended article: Characterization for certain class of non-symmetric skew lattices
description: scientific article; zbMATH DE number 836804