Let \(U\subseteq E(S)\) be a non-empty subset of idempotents of a semigroup \(S\). Define a relation \(\widetilde{\mathcal L}^U\) on \(S\) by setting \((a,b)\in \widetilde{\mathcal L}^U\) if and only if \(\{u\in U\mid au=a\}=\{u\in U\mid bu=b\}\). The relation \(\widetilde{\mathcal R}^U\) is defined dually. A semigroup \(S\) is called ‘\(U\)-semiabundant’ and denoted \((S,U)\) if any \(\widetilde{\mathcal L}^U\)-class and any \(\widetilde{\mathcal R}^U\)-class of \(S\) contain idempotents of \(U\) (a typical idempotent in the class containing element \(a\) is denoted by \(a^*\) and \(a^+\), respectively). A ‘\(U\)-semiabundant’ semigroup \((S,U)\) is called ‘\(U^\sigma\)-abundant’ if \(\widetilde{\mathcal L}^U\) is a right congruence, \(\widetilde{\mathcal R}^U\) is a left congruence on \((S,U)\) and \(x_1x_2\cdots x_n=x_{\alpha(1)}x_{\alpha(2)}\cdots x_{\alpha(n)}\) for any finite subset \(\{x_1,x_2,\dots,x_n\}\subseteq U\) and for any permutation \(\alpha\) on \(n\) letters.
Let \(T(Y)\) be an Ehresmann semigroup with \(Y\) a subsemilattice, and let \(L=\mathcal S(Y;L_\alpha;\varphi_{\alpha,\beta})\) and \(R=\mathcal S(Y;R_\alpha;\psi_{\alpha,\beta})\) be a strong semilattice of left zero bands \(L_\alpha\) and of right zero bands \(R_\alpha\), respectively.
The authors define the ‘quasi-spined product’ \(Q(L,T(Y),R;Y)\) of \(L\), \(T(Y)\) and \(R\), and prove that a semigroup \((S,U)\) is a \(U^\sigma\)-abundant semigroup if and only if \((S,U)\) is isomorphic to the quasi-spined product \(Q(L,T(Y),R;Y)\), where \(L\) is a left normal band, \(T(Y)\) is an Ehresmann semigroup and \(R\) is a right normal band. It turns out that the category of \(U^\sigma\)-abundant semigroups \((S,U)\) with admissible homomorphisms is isomorphic to the category of Ehresmann semigroups \((S,U)/\delta\) with admissible homomorphisms, where \(\delta\) is the minimum Ehresmann congruence on \((S,U)\).