Invertible elements of semirelatives and relatives (Q920128)

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.