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Special classes of positive implication algebras. (English) Zbl 1040.03049

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
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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
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