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On the Trivial Many Sorted Algebras and Many Sorted Congruences.pdf

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On the Trivial Many Sorted Algebras and Many Sorted Congruences

JOURNAL OF FORMALIZED MATHEMATICS Volume 8, Released 1996, Published 2002 Inst. of Computer Science, Univ. of Bia lystok On the Trivial Many Sorted Algebras and Many Sorted Congruences Artur Korni lowicz Institute of Mathematics Warsaw University Bia lystok Summary. This paper contains properties of many sorted functions between two many sorted sets. Other theorems describe trivial many sorted algebras. In the last section there are theorems about many sorted congruences, which are defined on many sorted algebras. I have also proved facts about natural epimorphism. MML Identifier: MSUALG_9. WWW: /JFM/Vol8/msualg_9.html The articles [29], [35], [9], [34], [1], [36], [38], [25], [37], [7], [27], [8], [4], [10], [30], [11], [2], [33], [28], [3], [5], [31], [32], [6], [12], [19], [26], [21], [23], [24], [20], [13], [15], [16], [14], [18], [17], and [22] provide the notation and terminology for this paper. 1. Preliminaries In this paper a, I denote sets and S denotes a non empty non void many sorted signature. The scheme MSSExD deals with a non empty set A and a binary predicate P, and states that: There exists a many sorted set f indexed by A such that for every element i of A holds P[i, f(i)] provided the following condition is met: ? For every element i of A there exists a set j such that P[i, j]. Let I be a set and let M be a many sorted set indexed by I. Observe that there exists an element of Bool(M) which is locally-finite. Let I be a set and let M be a non-empty many sorted set indexed by I. Observe that there exists a many sorted subset indexed by M which is non-empty and locally-finite. Let S be a non empty non void many sorted signature, let A be a non-empty algebra over S, and let o be an operation symbol of S. Note that every element of Args(o,A) is finite sequence-like. Let S be a non void non empty many sorted signature, let I be a set, let s be a sort symbol of S, and let F be an algebra family of I over S. Observe that every element of (SORTS(F ))(s) is