## 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:

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