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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
08A05 Structure theory of algebraic structures
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