Spined products of some semigroups (Q1327785)

Spined products of semigroups were first introduced and studied by \textit{N. Kimura} [Pac. J. Math. 8, 257-275 (1958; Zbl 0084.027)], and the spined product of a band and a special semigroup with respect to their common semilattice homomorphic image has been investigated by many authors. Let \(B\) be a band. For each \(i \in B\), let \(S_ i\) and \(D_ i\) be a semigroup and an oversemigroup of \(S_ i\) respectively such that \(D_ i \cap D_ j =\square\) for \(i \neq j\). Let \(\phi_{i,j}\) be a mapping of \(S_ i\) into \(D_ j\), and suppose that the family of \(\phi_{i,j}\) satisfies: (1) \(\phi_{i,i}\) is the identity mapping on \(S_ i\) for \(i \in B\); (2) \((S_ i \phi_{i, ij})\) \((S_ j \phi_{j,ij})\subset S_{ij}\), \(i, j \in B\); and (3) \([(a \phi_{i,ij}) (b \phi_{j,ij})] \phi_{ij,k} = (a \phi_{i,k}) (b\phi_{j,k})\) for \(a \in S_ i\), \(b \in S_ j\), \([ij] \geq [k]\), \(i,j,k \in B\), where \([s]\) denotes the class containing \(s\) in the greatest semilattice decomposition of \(B\). Then, \(S = \bigcup \{S_ i : i \in B\}\) becomes a band \(B\) of semigroups \(\{S_ i : i \in B\}\) under the multiplication \(*\) defined by \(a*b = (a\phi_{i, ij}) (b\phi_{j, ij})\). This \(S\) is denoted \(S = (B; S_ i, \phi_{i,j}, D_ i)\). In this paper, spined products are considered in connection with some special types of bands of semigroups. The authors firstly give a necessary and sufficient condition for a semigroup \(S\), which is a band \(B\) of semigroups \(\{S_ i : i \in B\}\), in order that \(S = \{B; S_ i, \phi_{i,j}, D_ i)\) for some \(D_ i\), \(\phi_{i,j}\), and secondly present a general composition for bands of semigroups that are (punched) spined products of a band and a semilattice of semigroups.