Algebraic Functions and EFD-sentences

CAMPERCHOLI MIGUEL; VAGGIONE DIEGO

Las Cruces, NM, EEUU

Congreso; BLAST; 2009

    An Equational Function Definition sentence (EFD-sentence for brevity) in the algebraic language L is a sentence of the form    ϕ=∀x₁,...,x_{n}∃!z₁,...,z_{m}⋀_{i=1}^{k}s_{i}(x,z)=t_{i}(x,z)where p_{i}, q_{i} are L-terms, n≥0 and m≥1. When an algebra A⊨ϕ, we can implicitly define m functions [ϕ]₁^{A},...,[ϕ]_{m}^{A}:Aⁿ→A by the system of equations    ⋀_{i=1}^{k}s_{i}(x,[ϕ]₁^{A}(x),...,[ϕ]_{m}^{A}(x))=t_{i}(x,[ϕ]₁^{A}(x),...,[ϕ]_{m}^{A}(x)).A function f is algebraic on A provided there is an EFD-sentence ϕ such that A⊨ϕ y f=[ϕ]_{j}^{A}. The algebraic functions on A form a clone that contains the clone of term-operations of A. Let Clo_{alg}(A) denote this clone. A natural problem is:<problem/><label>1</label>Given A characterize Clo_{alg}(A).    We show that Problem <ref>1</ref> is solved for every algebra in a variety V if we can solve the following related problem:<problem/><label>2</label>Describe the complete lattice of subclasses of V axiomatizable by EFD-sentences.    In recent years we have developed some tools that are useful in addresing these problems. A sizeable portion of our work on Problem <ref>2</ref> can be found in [1]. To mention an example, let D₀₁ be the variey of bounded distributive lattices. We have:<proposition/>(1) The subclasses of D₀₁ axiomatizable by EFD-sentences are the {trivial algebras}, {complemented distributive lattices} and D₀₁.(2) Let L∈D₀₁, then either L is complemented and Clo_{alg}(L)={boolean term functions on L}, or L is no complemented and Clo_{alg}(L)=Clo(L).    In our talk we give a survey of the main results and current state of our research on these topics.        References    [1] M. CAMPERCHOLI and D. VAGGIONE, Algebraically Expandlable Classes, to appear shortly in Algebra Universalis.