Recent developments in the theory of skew lattices. (English) Zbl 0844.06003

Leech, Jonathan

Semigroup Forum 52, No. 1, 7-24 (1996).

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.

Reviewer: G.A.Grätzer (Winnipeg)

MSC:

06B05	Structure theory of lattices
20M99	Semigroups
06-02	Research exposition (monographs, survey articles) pertaining to ordered structures

Keywords:

survey; skew lattice; rectangular band

References:

[1]	Bignall, R. J.,Quasiprimal varieties and components of universal algebras, Dissertation, The Flinders University of S. Australia, 1976.
[2]	Bignall, R. J.,Non-commutative multiple-valued logic, Proceedings of the Twenty-first International Symposium on Multiple-Valued Logic, IEEE Computer Society Press (1991), 49–54.
[3]	Bignall, R. J., and J. E. Leech,Skew Boolean algebras and discriminator varieties, to appear in A. Universalis. · Zbl 0821.06013
[4]	Burris, S., and H. P. Sankappanavar, ”A Course in Universal Algebra,” Springer-Verlag, New York, 1981. · Zbl 0478.08001
[5]	Cornish, W. H.,Boolean skew algebras, Acta Math. Hungar.36 (1980), 281–291. · Zbl 0465.06010
[6]	Gerhardts, M. D.,Zur Charakterisierung distributiver Schiefverbände, Math. Ann.161 (1965), 231–240. · Zbl 0151.01701
[7]	–,Zerlegungshomomorphismen in Schrägverbänden, Arch. Math.21 (1970) 116–122. · Zbl 0199.32201
[8]	Howie, J. M., ”Introduction to Semigroup Theory,” Academic Press, London, 1976. · Zbl 0355.20056
[9]	Jordan P.,Halbgruppen von Idempotenten und nichtkommutative Verbände, J. Reine Angew. Math.211 (1962), 136–161. · Zbl 0118.02303
[10]	Keimel, K., and H. Werner,Stone duality for varieties generated by quasiprimal algebras, Mem. Amer. Math. Soc.148 (1974), 59–85. · Zbl 0283.08001
[11]	Kimura, N.,The structure of idempotent semigroups, I, Pacific J. Math.8 (1958), 257–275. · Zbl 0084.02702
[12]	Leech, J. E.,Skew lattices in rings, A. Universalis26 (1989), 48–72. · Zbl 0669.06006
[13]	–,Skew Boolean algebras, A. Universalis27 (1990), 497–506. · Zbl 0719.06010
[14]	–,Normal skew lattices, Semigroup Forum44 (1992), 1–8. · Zbl 0754.06004
[15]	–,The geometric structure of skew lattices, Trans. Amer. Math. Soc.335 (1993), 823–842. · Zbl 0792.06008
[16]	Matsushita, S.,Zur Theorie der nichtkommutativen Verbände, I, Math. Ann.137 (1959), 1–8. · Zbl 0083.02405
[17]	Pastjin, F., and A. Romanowska,Idempotent distributive semirings, I, Acta Sci. Math.44 (1982), 239–253. · Zbl 0514.16026
[18]	–,Idempotent distributive semirings, II, Semigroup Forum26 (1983), 151–166. · Zbl 0514.16027
[19]	Petrich, M. ”Lectures on Semigroups,” John Wiley and Sons, New York, 1977. · Zbl 0369.20036
[20]	Rosenthal, K. I., ”Quantales and their Applications,” Longman Scientific and Technical, Essex, 1990. · Zbl 0703.06007
[21]	Schein, B. M.,Restrictive bisemigroups, Amer. Math. Soc. Trans. (2)100 (1972), 293–307. · Zbl 0247.20066
[22]	–,Pseudosemilattices and pseudolattices, Amer. Math. Soc. Transl. (2)119 (1983), 1–16. · Zbl 0502.06001
[23]	Schweigert, D.,Near lattices, Math. Solvaca35 (1985), 313–317. · Zbl 0578.06005
[24]	–,Distributive associative near lattices, Math. Slovaca35 (1985), 313–317. · Zbl 0578.06005