Algebra of Morphisms
نویسنده
چکیده
Let I be a set and let A, f be functions. The functor f↾IA yielding a many sorted function indexed by I is defined by: (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 idUnionA↾IA = idA. (2) Let I be a set, A, B be many sorted sets indexed by I, and f , g be functions. If rngκ(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 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 if and only if len p = i. (6) For every set A and for every natural number i holds A ⊆ A∗. (7) For every set A and for every natural number i holds i 6= 0 and A = ∅ iff A = ∅.
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