Transformations Between Menger Systems

Abstract

Abstract. To define transformations between based universal algebras we must introduce representations that depend on the bases, contrary to what was possible for general vector spaces and believed possible for universal algebras. In fact, a counterexample shows that by representation-free transformations alone one cannot even ascertain whether a universal algebra has any dimension or not.A transformation notion, which can do, concerns basis dependent Menger systems. It enjoys a basic geometric property of universal algebras, the preservation of reference flocks, and generalizes the transformation groups of Linear Algebra into groupoids. 0. Preliminaries 0.0. Introduction. The necessity of a transformation notion, distinct from isomorphisms, was acknowledged in vector spaces since 1889 [13]. In Universal Algebra, on the contrary, no notion of a transformation appeared, just some isomorphism variants (equivalence between algebras [3] and Marczewski's weak or general [1] isomorphisms) did. Since till last year even simple definitions of vector spaces as universal algebras [11,12] lacked, this made conceivable that their two fields are distinct.After introducing some notions of "Universal Mathematics", this paper provides universal algebras with a candidate for a transformation notion together with a counterexample to the belief that isomorphism ideas suffice. Its continuation (to appear here under the title "Sameness between based universal algebras") will validate this candidate by proving its equivalence to other new notions. Other motivations are in [10]. 0.1. Notation. We conform to [4], but for the following few differences. We denote the set-theoretical pair {{a}, {a, 6}} by (a, b), yet we still simplify /((a, b)) into f{a,b) and ({x,y),z) into (x,y,x) as in [4]. PX denotes the set of subsets of set X and %x its identity function. We consider functional composition as the restriction of relational composition, here denoted by • , namely f g is "the composition of g and /" and (/ " 9)( x ) = f(9{ x ))-Accordingly, we perform the restriction of a function / to some set S merely by functional composition: f • isAs usual, we write f: A^B to say that / is a function with arguments in the whole set A and values in B, f: A^B or /: A-*~B to say that it also is one to one or onto B and /: An-^-B to say it is a bijection onto B. We will forget that "function -domain" and "family -index" are pairwise synonymic and we avoid the notation {aj}je/ or (<Zj | i G I). Within informal comments we will replace "function" with "indexing", to emphasize values. Also, we denote the set-theoretical power A B = {/ | f:A -> B} as the arithmetic one (The latter will not occur here.) Endomorphism representations.Let Ea C A A be the set of all endomorphisms of an algebra a on A. Given a set X, let b :X -> A and consider the function rh:When a function b:X -> A serves to define such a sampling of endomorphisms, we call it a frame of a. If this sampling represents every endomorphism by any sample and conversely, namely if we get that ...