Skew lattices in rings. (English) Zbl 0669.06006

Leech, Jonathan

Algebra Univers. 26, No. 1, 48-72 (1989).

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

Keywords:

skew lattices in rings; noncommutative lattice; rectangular skew lattice; maximal rectangular subalgebras; congruence relation; Artinian rings

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