Stratisfied semigroups
A semigroup \(S\) with zero 0 is called stratisfied if \(\bigcap_{m > 0} S^ m = \{0\}\). (The definition is effective for any semigroup, that is, a semigroup \(S\) is stratisfied if \(S^ 0\) is stratisfied.) The depth function \(\lambda : S \setminus \{0\} \to N\) assigns to each \(s \in S\), \(s\neq 0\), the greatest positive integer \(m\) such that \(s \in S^ m\). A semigroup \(S\) is called homogeneous if it has a presentation in which the defining relations are of the form either \(u = 0\) or \(u = v\) with words \(u\) and \(v\) of equal lengths. A product \(a_ 1 a_ 2 \cdots a_ r\) is called proper if \(a_ 1 a_ 2 \cdots a_ r \neq 0\) and \(\lambda(a_ 1 a_ 2 \cdots a_ r) = \lambda a_ 1 + \lambda a_ 2 + \cdots + \lambda a_ r\). Then \(S\) is homogeneous if and only if \(S\) is stratisfied and all non-zero products in \(S\) are proper. The depth of a subset \(C \supsetneqq \{0\}\) of \(S\) is defined by \(\mu(C) = \max \{\lambda a: a\neq 0,\;a \in C\}\). The following are basic principles of this paper. (1) Let \(\rho\) be a congruence on a stratisfied semigroup. \(S/\rho\) is stratisfied if and only if every non-zero \(\rho\)- class has finite depth (the depth of a non-zero \(\rho\)-class \(C\) in \(S/\rho\) coincides with its depth as a subset of \(S\)). (2) Let \(S\) be homogeneous. If \(\rho\) is homogeneous, that is, every non-zero \(\rho\)- class is contained in a single layer (i.e. one of \(S^ m\setminus S^{m+1}\)), then \(S/ \rho\) is homogeneous. If \(S/ \rho\) is homogeneous and the \(\rho\)-class of every \(x \in S\setminus S^ 2\) has depth 1 then \(\rho\) is homogeneous. The class of stratisfied semigroups contains free semigroups, free commutative semigroups, homogeneous semigroups and nilpotent semigroups. Let \(\mathcal V\) be the variety containing all commutative semigroups. Let \(F\) be a free \(\mathcal V\)-semigroup and consider a homogeneous equivalence relation (h.e.r.) of degree \(m\) on \(F\), that is, an equivalence relation on \((F^ m \setminus F^{m+1}) \cup \{0\}\). Every stratisfied \(\mathcal V\)- semigroup determines two infinite sequences (lower trace and upper trace) of h.e.r. on a free \(\mathcal V\)-semigroup called traces. If \(S\) is homogeneous, then the two traces coincide and \(S\) is determined by the trace. If \(S\) is stratisfied, the traces produce two homogeneous semigroups. In other words those semigroups can be constructed by h.e.r. For example, homogeneous semigroups and homogeneous nilpotent semigroups. Reviewer's remark. This paper extensively develops the reviewer's work [Osaka Math. J. 10, 191-204 (1958; Zbl 0084.026)].
scientific article; zbMATH DE number 716384