On a class of semigroups with symmetric presentations
The authors study semigroups \(S(m,n)\) with the presentations \[ \langle x_ 1,\dots, x_ n:\;x_ i^{m+1}= x_ i,\;x_ i x_ j^ 2= x_ j x_ i^ 2,\;1\leq i,j\leq n\rangle. \] The structure of these semigroups is determined as follows: if \(m\) is odd then \(S(m,n)\) is a semilattice of groups, if \(m\) is even then \(S(m,n)\) is a semilattice of right groups. For those \((m,n)\) for which \(S(m,n)\) is finite, formulas for the cardinality \(| S(m,n)|\) are obtained. For odd \(m\) this is the case iff \((m,3)=1\) or \((m,3)=3\) and \(n\leq 3\); for even \(m\) this happens iff \(m\leq 4\).
scientific article; zbMATH DE number 164968