Extensions of the Positive Closure Operator by Using Logical Connectives

Abstract

The positive closure operator is defined on using the logical formulas containing the logical connectives ∨, & and the quantifier ∃. Extensions of the positive closure operator are considered by using arbitrary (and not necessarily binary) logical connectives. It is proved that each proper extension of the positive closure operator by using local connectives gives either an operator with a full systemof logical connectives or an implication closure operator (extension by using logical implication). For the implication closure operator, the description of all closed classes is found in terms of endomorphism semigroups.