Abstract
The aim of this article is to provide structure theorems for weakly B-abundant semigroups satisfying the Congruence Condition (C), where B is a band. Such semigroups may be thought of as generalisations of orthodox semigroups. Our focus is on providing a description of a weakly B-abundant semigroup S with (C) as a spined product of a weakly B-abundant semigroup SB (depending only on B) and S/γB, where γB is the analogue of the least inverse congruence on an orthodox semigroup. This result is an analogue of the Hall-Yamada theorem for orthodox semigroups. In the case that B is a normal band, or S is weakly B-superabundant, we find a closed form for γB. In addition, we build on an existing result of Ren to show that a weakly B-superabundant semigroup (C) is a semilattice of rectangular bands of monoids.
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