Sequential convergences in dually residuated lattice ordered semigroups (Q1907288)

Sequential convergences in \(l\)-groups and in Boolean algebras were studied by J. Jakubík and M. Harminc (see below). A quadruple \((A, +, \leq, -)\) is said to be a dually residuated lattice-ordered semigroup (shortly DRL-semigroup) if (i) \((A, +, \leq)\) is a commutative lattice-ordered semigroup with identity 0, (ii) for every \(a,b \in A\) there exists a least \(x \in A\) (denoted by \(a - b)\) such that \(b + x \geq a\), (iii) \((a - b) \vee 0 + b \leq a \vee b\) for \(a,b \in A\) and \(a - a \geq 0\) for all \(a \in A\). DRL-semigroups contain both Boolean algebras and \(l\)-groups as special cases but also other algebraic systems. In the paper under review, the author introduces and investigates a notion of sequential convergence in a DRL-semigroup. She obtains results analogous to those of \textit{J. Jakubík} [Czech. Math. J. 38(113), No. 3, 520--530 (1988; Zbl 0668.54002); Math. Slovaca 38, No. 3, 269--272 (1988; Zbl 0662.06005)], the reviewer [in: Convergence structures, Proc. Conf., Bechyně/Czech. 1984, Math. Res. 24, 153--158 (1985; Zbl 0581.06009); Czech. Math. J. 37(112), 533--546 (1987; Zbl 0645.06006); ibid. 39(114), No. 2, 232--238 (1989; Zbl 0681.06007)], and the reviewer and \textit{J. Jakubík} [ibid. 39(114), No. 4, 631--640 (1989; Zbl 0703.06011)].