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\(L\)-ordered and \( L\)-lattice ordered groups. (English) Zbl 1387.06005
Summary: This paper pursues an investigation on groups equipped with an \( L\)-ordered relation, where \( L\) is a fixed complete Heyting algebra. First, by the concept of join and meet on an \( L\)-ordered set, the notion of an \( L\)-lattice (a weak \( L\)-lattice) is introduced and some related results are obtained. Then we applied them to define an \( L\)-lattice ordered group. We also introduce convex \( L\)-subgroups to construct a quotient \( L\)-ordered group. At last, a relation between the positive cone of an \( L\)-ordered group and special type of elements of \(L^G\) is found, where \(G\) is a group.

MSC:
06A75 Generalizations of ordered sets
06D20 Heyting algebras (lattice-theoretic aspects)
06F15 Ordered groups
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