Heterogeneous Algebras

Birkhoff-Frink representations as functors

“…In all that follows we use standard concepts and constructions from category theory, see e.g., [1], [17], [18], [19], and [22]; classical universal algebra, see e.g., [6], [10], [16], [21], and [25]; many-sorted algebra, see e.g., [2], [4], [20], [24], and [28]; and lattice theory, see e.g., [3], [6], and [25]. However, following the French mathematical tradition, we agree to call a functor F : C / / D essentially surjective (instead of representative or isomorphism-dense) if for every object d in D there exists an object c in C such that F (c) is isomorphic to d.…”

On elementary equivalence and isomorphism of clone segments

“…. , CO} is a function for which (l), (2) and (3) for all (a, b) E C, then all these metric spaces can be constructed over the same underlying set.…”

“…Clones of topological spaces and their first order language were extensively investigated in [9]. T o d escribe this language effectively, it is useful to view a clone as an w-sorted algebra (where w denotes the set of all finite ordinals), a concept introduced in [2]. In this view, a clone is an w-sorted algebra such that (n) (n) (c) there are n nullary operations co , .…”

“…Proof of Theorem 2. Let C be a set and let 'p : C x C + { 1,2, , oo} be a function satisfying (l),(2) and(3). For any m E w, write Em = {(a,b) E C x C ] cp(a, b) > m}.…”

Nonlinear matrix decompositions and an application to parallel processing

“…The minimax algebra, a heterogeneous algebra in the sense of Birkhoff [14], is a nonlinear algebraic structure whose underlying value set is a bounded lattice-ordered group. Because algebraic properties of the bounded lattice-ordered group F induce properties on the set of matrices having values in IF, the matrix calculus relies heavily on specific algebraic attributes of F. Note that this also occurs in linear algebra, in which properties of the vector space of all m x n matrices over the field of complex numbers C are intimately related to the algebraic properties of C. In addition, the analysis and continued investigation of linear transforms in applications is greatly facilitated by theoretical results of linear algebra.…”