Subalgebras of Many Sorted Algebra. Lattice of Subalgebras
Ewa Burakowska · 1996
In this paper x will be arbitrary. The scheme LambdaB concerns a non empty set A and a unary functor F yielding arbitrary, and states that: There exists a function f such that dom f = A and for every element d of A holds f(d) = F(d) for all values of the parameters. Let I be a set, let X be a many sorted set of I, and let Y be a non-empty many sorted set of I. Observe that X∪Y is non-empty and Y ∪X is non-empty. Next we state two propositions: (1) Let I be a set, and let X be a many sorted set of I, and let Y be a non-empty many sorted set of I. Then X ∪ Y is non-empty and Y ∪X is non-empty. (2) For every non empty set I and for all many sorted sets X, Y of I and for every element i of I∗ holds ∏ ((X ∩ Y ) · i) = ∏ (X · i) ∩ ∏ (Y · i). Let I be a set and let M be a many sorted set of I. A many sorted set of I is said to be a many sorted subset of M if: (Def.1) It ⊆ M. Let I be a set and let M be a non-empty many sorted set of I. Observe that there exists a many sorted subset of M which is non-empty.