On the regularity of the elements of a relative. (Russian) Zbl 0714.06006

Skornyakov, L. A.

Arch. Math., Brno 25, No. 1-2, 103-106 (1989).

Let \((R;+,\cdot,-,0,1)\) be a Boolean algebra and let (R;\(\circ,*,E)\) be a monoid with an involution * such that the relations \((x+y)^*=x^*+y^*\), \((x+y)\circ z=x\circ z+y\circ z\) and \(x^*\circ \overline{x\circ y}\leq \bar y\) hold for every x,y,z\(\in R\). Then the algebra \((R;,+,\cdot,-,0,1,\circ,*,E)\) is said to be a relative. An element \(\mu\) is said to be regular if and only if \(\mu =\mu \circ r\circ \mu\) for some \(r\in R\). In this note some characterizations of regular elements of a relative R are proved. In particular, it is shown that \(\mu\) is regular iff \(\mu\leq \mu \circ \overline{\mu^*\circ {\bar \mu}\circ \mu^*}\circ \mu\) and iff \(\mu =\mu \circ \overline{\mu^*\circ {\bar \mu}\circ \mu^*}\circ \mu\).

Reviewer: E.Płonka

MSC:

06E99	Boolean algebras (Boolean rings)
03E20	Other classical set theory (including functions, relations, and set algebra)
20M20	Semigroups of transformations, relations, partitions, etc.
03G15	Cylindric and polyadic algebras; relation algebras

Keywords:

relation algebra; Boolean algebra; relative

Citations:

Zbl 0674.06012