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Invertible elements of semirelatives and relatives. (English. Russian original) Zbl 0708.08001
Mosc. Univ. Math. Bull. 44, No. 2, 55-58 (1989); translation from Vestn. Mosk. Univ., Ser. I 1989, No. 2, 37-40 (1989).
A semirelative is by definition a universal algebra S of the signature $$\{+,\cdot,\circ,I,0,*,E\},$$ where $$+$$, $$\cdot$$ and $$\circ$$ are binary operations, * is unary and 0, I, E are 0-ary operations, such that:
(S1) $$\{S| +,\cdot,0,I\}$$ is a distributive lattice with zero and unit,
(S2) $$\{$$ $$S| \circ,*,E\}$$ is a monoid with involution,
and for any a,b,c,d$$\in S$$ holds:
(S3) $$(a+b)\circ c=a\circ c+b\circ c,$$
(S4) $$(a+b)^*=a^*+b^*,$$
(S5) $$(a\circ b)\cdot (c\circ d)\leq a\circ ((a^*\circ c)\cdot (b\circ d^*))\circ d,$$
(S6) $$I\circ 0=0.$$
If one replaces (S1) by:
(R1) $$\{S| +,\cdot,0,I\}$$ is a Boolean algebra,
and (S5), (S6) to:
(R5) $$a^*\circ a\circ b\leq b,$$
then one gets the definition of a relative, which is a special case of a semirelative.
It is proved that 1) an element a of a semirelative S is invertible (i.e. $$a\circ b=b\circ a=E$$ for some $$b\in E)$$ iff $$a\circ a^*=a^*\circ a=E$$; 2) an element a of a relative R is invertible iff a is a maximal element of the sets $${\mathfrak A}=\{x\in R|$$ $$x\circ x^*\leq E\}$$ and $${\mathfrak B}=\{y\in R|$$ $$y^*\circ y\leq E\}.$$
Besides, an example of a semirelative not satisfying 2) is constructed.
Reviewer: I.Shestakov
##### MSC:
 08A05 Structure theory of algebraic structures
semirelative