Implicative partially ordered semigroups (Q5929669)

A partially ordered semigroup \((S,\cdot,\leq)\) is called implicative if \((S,\cdot)\) has an identity which is maximal in \((S,\leq)\), and if for all \(x,y\in S\) the set \(\{z\in S\mid zx\leq y\}\) has a greatest element denoted by \(x * y\). This concept generalizes the notion of implicative semilattice [\textit{W. C. Nemitz}, ``Implicative semi-lattices'', Trans. Am. Math. Soc. 117, 128-142 (1965; Zbl 0128.24804)]. Several properties of such semigroups are derived and the concept of a strong filter is introduced. For any implicative partially ordered semigroup it is shown that (1) if \(F\) is a strong filter of \(S\) then the relation: \(xR_Fy\) if and only if \(cx\leq y\) and \(dy\leq x\) for some \(c,d\in F\), is a congruence on \((S,\cdot,*)\); (2) a binary relation \(R\) on \(S\) is a convex congruence on \(S\) if and only if \(F_1=\{x\in S\mid xR1\}\) is a strong filter of \(S\) and \(R=R_{F_1}\); (3) if a strong filter \(F\) satisfies also \(x*y\in F\) for all \(x,y\in F\), then \(S/R_F\) forms an implicative partially ordered semigroup with respect to: \(F_x\preceq F_y\) if and only if \(cx\leq y\) for some \(c\in F\). The main result of the paper states that every implicative partially ordered semigroup \(S\) is isomorphic with the direct product \(N_X\times T_S\), if \(N_S\) denotes the set \(\{x\in S\mid x\leq 1\}\) and if \(T_S=\{x\in S\mid x\) is maximal in \((S,\leq)\}\) is closed with respect to the operation \(*\). NEWLINENEWLINENEWLINEReviewer's remark. The reason why the concept of a residual (well established in the literature [see: \textit{L. Fuchs}, Partially ordered algebraic systems. Oxford etc.: Pergamon Press (1963; Zbl 0137.02001); or \textit{T. S. Blyth} and \textit{M. Janowitz}, Residuation theory. Oxford etc.: Pergamon Press (1972; Zbl 0301.06001)]) is avoided, is not clear. The origin in the theory of implicative semilattices would also allow a generalization in this direction: \(x*y=\max\{z\in S\mid zy\leq x\}\).