Closure Operators and Subalgebras 1 MML Identifier: WAYBEL10.

[9] provide the notation and terminology for this paper. 1. PRELIMINARIES In this article we present several logical schemes. The scheme SubrelstrEx deals with 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], and • 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], and • For all elements a, b of B holds a ≤ b iff Q [a,b]. The scheme SubrelstrEq1 deals with a non empty relational structure A, non empty full relational substructures B, C of A, and a unary predicate P, and states that: The relational structure of B = the relational structure of C provided the following conditions are met: • For every set x holds x is an element of B iff P [x], and • For every set x holds x is an element of C iff P [x]. The scheme SubrelstrEq2 deals with a non empty relational structure A, non empty full relational substructures B, C of A, and a unary predicate P, and states that: The relational structure of B = the relational structure of C provided the parameters have the following properties: • For every element x of A holds x is an element of B iff P [x], and • For every element x of A holds x is an element of C iff P [x]. Next we state three propositions