Communications of the
Korean Mathematical Society CKMS

The concepts of $*$-relation of a ($\Gamma$-)semigroup and ${\bf\bar{\Gamma}}$-adequate transversal of a ($\Gamma$-)abundant semigroup are defined in this note. Then we develop a matrix type theory for abundant semigroups. We give some equivalent conditions of a Rees matrix semigroup being abundant and some equivalent conditions of an abundant Rees matrix semigroup having an adequate transversal. Then we obtain some Rees matrix representations for abundant semigroups with adequate transversals by the above theories.

Keywords: abundant semigroup, adequate semigroup, $\Gamma$-adequate transversal

MSC numbers: 20M10