Weakly abundant semigroups within certain subsemigroups of the semigroup of full contraction mappings of a finite chain

Let [Formula: see text] be a finite n-chain [Formula: see text], and let [Formula: see text] be the monoid of full contraction mappings on [Formula: see text]. Denote by [Formula: see text] and [Formula: see text] to be the monoid of order-decreasing full contractions, the monoid of all order-preserving full contractions, the monoid of all order-preserving or order-reversing full contractions, and the monoid of all order-decreasing and order-preserving full contraction mappings on [Formula: see text], respectively. In this paper, we characterize the generalized starred Green’s relations [Formula: see text] and [Formula: see text] on every monoid [Formula: see text], and show that for any monoid [Formula: see text], S is both weakly abundant and weakly [Formula: see text]-unipotent, while for any monoid [Formula: see text], it is neither. This further demonstrates that the decreasing property affects the weakly abundance property on any [Formula: see text].