Jun, Young Bae; Kim, Hee Sik; Ahn, Sun Shin Special classes of positive implication algebras. (English) Zbl 1040.03049 Inf. Sci. 152, 203-215 (2003). Summary: We construct a positive implication algebra PI(\(A\)) inherited from a chain. For a non-empty subset \(\nabla\) of a positive implication algebra, using the special set \(\nabla (a,b)\), we give a necessary and sufficient condition for \(\nabla\) to be an implicative filter. We also introduce the notion of a \(\nabla\)-type positive implication algebra, and prove the followings: (1) every \(\nabla\)-type positive implication algebra is a commutative monoid under the operation \(\odot\), but not a group; (2) every \(\nabla\)-type positive implication algebra is a lower semilattice; (3) in a \(\nabla\)-type positive implication algebra the class of implicative filters coincides with the class of convex subsemigroups with \(V\). Finally we show that every \(\nabla\)-type positive implication algebra is a semi-Brouwerian algebra, and consider the direct product of implicative algebras. MSC: 03G25 Other algebras related to logic 06A12 Semilattices 20M10 General structure theory for semigroups Keywords:Positive implication algebra inherited from a chain; Implicative filter; Special implicative filter; implicative filter; Lower semilattice; \(\nabla\)-type positive implication algebra; Semi-Brouwerian algebra PDFBibTeX XMLCite \textit{Y. B. Jun} et al., Inf. Sci. 152, 203--215 (2003; Zbl 1040.03049) Full Text: DOI References: [1] Rasiowa, H., An algebraic approach to non-classical logics (1974), American Elsevier Publishing Co. Inc: American Elsevier Publishing Co. Inc New York · Zbl 0299.02069 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.