Abstract
We prove that for a finite first order structure {mathbf {A}} and a set of first order formulas Phi in its language with certain closure properties, the finitary relations on A that are definable via formulas in Phi are uniquely determined by those of arity |A|^{2}. This yields new proofs for some finiteness results from universal algebraic geometry.
Highlights
To every algebraic structure A, one can associate certain subsets of its finite direct powers An (n ∈ N)
Two algebras defined on the same base set A, but possibly with different basic operations, may have the same relational clone: if A is finite, this happens if and only if the two algebras are term equivalent, i.e., each fundamental operation of one algebra is a term operation of the other algebra [8, p. 55, Folgerung 1.2.4]
Rossi inequivalent equational domains [6, Theorem 3]. We observe that this finiteness can be obtained by considering the clonoid of the characteristic functions of algebraic sets and applying a consequence of the Baker Pixley Theorem [2, Theorem 2.1] that was recently proved by Sparks [9, Theorem 2.1] to these clonoids
Summary
To every algebraic structure A, one can associate certain subsets of its finite direct powers An (n ∈ N). Two algebras defined on the same base set A, but possibly with different basic operations, may have the same relational clone: if A is finite, this happens if and only if the two algebras are term equivalent, i.e., each fundamental operation of one algebra is a term operation of the other algebra [8, p. An) for all i ∈ I}, where fi, gi are term operations from A We will call such a solution set S an algebraic set. Using a description of algebraic equivalence through certain invariants, Pinus proved that on a finite set, there are at most finitely many algebraically.
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