Closure Operators and Subalgebras

In this article we present several logical schemes. The scheme SubrelstrEx concerns a non empty relational structure A, a set B, and a unary predicate P, and states that: There exists a non empty full strict relational substructure S of A such that for every element x of A holds x is an element of S if and only if P[x] provided the following conditions are met: • P[B], • B ∈ the carrier of A. The scheme RelstrEq deals with non empty relational structures A, B, a unary predicate P, and a binary predicate Q, and states that: The relational structure of A = the relational structure of B provided the following conditions are met: • For every set x holds x is an element of A iff P[x], • For every set x holds x is an element of B iff P[x], • For all elements a, b of A holds a ¬ b iff Q[a, b], • For all elements a, b of B holds a ¬ b iff Q[a, b].