@MISC{Bancerek_mmlidentifier:catalg_1., author = {Grzegorz Bancerek}, title = {MML Identifier:CATALG_1. Algebra of Morphisms}, year = {} }

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Abstract

Let I be a set and let A, f be functions. The functor f ↾ IA yields a many sorted function indexed by I and is defined as follows: (Def. 1) For every set i such that i ∈ I holds ( f ↾ IA)(i) = f↾A(i). One can prove the following propositions: (1) For every set I and for every many sorted set A indexed by I holds id � A ↾ IA = idA. (2) Let I be a set, A, B be many sorted sets indexed by I, and f, g be functions. If rng κ ( f ↾ IA)(κ) ⊆ B, then (g · f) ↾ IA = (g ↾ IB) ◦(f ↾ IA). (3) Let f be a function, I be a set, and A, B be many sorted sets indexed by I. Suppose that for every set i such that i ∈ I holds A(i) ⊆ dom f and f ◦ A(i) ⊆ B(i). Then f ↾ IA is a many sorted function from A into B. (4) Let A be a set, i be a natural number, and p be a finite sequence. Then p ∈ A i if and only if len p = i and rng p ⊆ A. (5) Let A be a set, i be a natural number, and p be a finite sequence of elements of A. Then p ∈ A i if and only if len p = i. (6) For every set A and for every natural number i holds A i ⊆ A ∗.