## Skew lattices in rings.(English)Zbl 0669.06006

A skew lattice is a noncommutative lattice satisfying the absorption laws: $$x\vee (x\wedge y)=x=x\wedge (x\vee y)$$, $$(x\wedge y)\vee y=y=(x\vee y)\wedge y$$. Let $$D=L\times R$$ be the Cartesian product of two nonempty sets. Then the operations $$(a,b)\vee (a',b')=(a',b)$$, $$(a,b)\wedge (a',b')=(a,b')$$ define on D a skew lattice, which is called rectangular skew lattice. The relation $$x\equiv y$$ iff $$x\wedge y\wedge x=x$$ and $$y\wedge x\wedge y=y$$ is a congruence relation of the skew lattice S, the equivalence classes are maximal rectangular subalgebras, and the factor is the maximal lattice image of S. x$${\mathcal R}y$$ iff $$x\wedge y=y$$ and $$y\vee x=x$$ or dually $$x\vee y=x$$ and $$y\vee x=y$$. Then $${\mathcal R}$$ is a congruence relation of S. Let $$(A,+,\cdot)$$ be a unitary ring, then two operations are defined on A: $$x\vee y=x+y-xy$$, $$x\wedge y=x\cdot y$$. E(A) denotes the set of all idempotents. Let $$S\subseteq E(A)$$ be a set closed under $$\vee$$ and $$\wedge$$, then (S,$$\vee,\wedge)$$ is a skew lattice in (A,$$\vee,\wedge)$$. Some properties of these skew lattices are established. Finally, the author considers skew lattices which correspond to simple and semisimple Artinian rings.
Reviewer: E.T.Schmidt

### MSC:

 06B99 Lattices
Full Text:

### References:

 [1] Dilworth, R. P.,Lattices with unique complements, Trans. AMS,57 (1945), pp. 123-54. · Zbl 0060.06103 [2] Gerhardts, M. D.,Zur Charakterisierung distributiver Schiefverb?nde, Math Ann.,161 (1965), pp. 231-240. · Zbl 0151.01701 [3] Gerhardts, M. D.,Schr?gverb?nde und Quasiordnungen, Math. Ann.,181 (1969), pp. 65-73. · Zbl 0175.01403 [4] Howie, J. M.,An Introduction to Semigroup Theory, Academic Press, 1976. · Zbl 0355.20056 [5] Jordan, P.,Uber nichtkommutative Verbande, Arch. Math.,2 (1949), pp. 56-59. · Zbl 0037.15604 [6] Jordan, P.,Halbgruppen von Idempotenten und nichtkommutative Verb?nde, J. reine angew. Math.,211 (1962), pp. 136-161. · Zbl 0118.02303 [7] Leech, J. E.,Normal Skew Lattices, Semigroup Forum, to appear. · Zbl 0754.06004 [8] Leech, J. E.,Primitive Skew Lattices and Completely Simple Semigroups, submitted. [9] Leech, J. E.,Skew Lattices of Partial Functions, Algebra Universalis, to appear. · Zbl 0792.06008 [10] Petrich, M.Lectures on Semigroups, John Wiley and Sons, 1977. · Zbl 0369.20036 [11] Schein, B.,Pseudosemilattices and Pseudolattices, Amer. Math. Soc. Transl. (2),119 (1983), pp. 1-16. · Zbl 0502.06001 [12] Schweigert, D.,Near Lattices, Math. Slovaca,23 (1982), pp. 313-317. · Zbl 0492.06007 [13] Schweigert, D.,Distributive Associative Near Lattices, Math. Slovaca,35 (1985), pp. 313-317. · Zbl 0578.06005
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