The translational hull of reductive matrix 0-bands

Abstract

We consider matrix 0-bands [ γ] 0 where γ is a dense relation on a set V. If the semigroup [γ] 0 is reductive, then every bitranslation of [ γ] 0 is in one-to-one correspondence with a left adjoint map on a so-called weakly ordered set ( W, ◁) with 0 and 1, which is closely related to ( V, γ). This interpretation of the translational hull ω([ γ] 0) of a reductive square matrix 0-band is useful for characterizing several semigroups of residuated mappings on bounded posets. In particular, some part of Zareckiǐ's work on binary relations can be seen in this light. This paper thus provides a link between the general results of Petrich on the tranlational hull of Rees matrix semigroups and Zareckiǐ's abstract characterization of certain semigroups of binary relations.