Permutative semigroups whose congruences form a chain.
A semigroup, whose congruences form a chain, is called a \(\Delta\)-semigroup. The commutative \(\Delta\)-semigroups were characterized by \textit{B. M. Schein} [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 17, 523-527 (1969; Zbl 0187.29103) and ibid. 23(1975), 1247-1248 (1976; Zbl 0329.20043)] and independently by \textit{T. Tamura} [Bull. Soc. Math. Fr. 97, 369-380 (1969; Zbl 0191.01705)]. A semigroup \(S\) is permutative if \(S\) satisfies an identity of the form: \(x_1x_2\cdots x_n=x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\sigma(n)}\) for some non-identity permutation \(\sigma\) of \(\{1,2,\dots,n\}\). In this paper the permutative \(\Delta\)-semigroups \(S\) are completely determined. If \(S\) is not commutative then surprisingly there are only six permutative \(\Delta\)-semigroups, each of which has at most three elements. For the proof three observations on a permutative semigroup \(S\) are essential: (1) \(S\) is a semilattice of Archimedean semigroups; (2) if \(S\) is a \(\Delta\)-semigroup then \(S\) is medial (i.e., \(S\) satisfies the special permutation identity: \(axyb=ayxb\)); (3) the medial \(\Delta\)-semigroups are known [\textit{P. G. Trotter}, Semigroup Forum 12, 313-331 (1976; Zbl 0338.20071)].
noncommutative permutative \(\Delta\)-semigroups
scientific article; zbMATH DE number 2138812