General Algebraic Geometry and Formal Concept Analysis

This paper describes the interaction between classical Algebraic Geometry, General Algebra, and Formal Concept Analysis. The mathematical foundations of Formal Concept Analysis can be found in ([GW99]). Its goal is to elaborate the general core of the basic results of Algebraic Geometry using this interaction. We start from a general polynomial context of the form Kn,A := (A, Fn(X,A) × Fn(X,A),⊥). Here A is a general algebra, Fn(X,A) is the free algebra in n variables in the variety VarA generated by A, and we have ~a ⊥ (p, q) : ⇐⇒ p(~a) = q(~a) for ~a ∈ A and p, q ∈ Fn(X,A). Extents of this formal context will be called A-algebraic sets. We find that the intents of K are certain congruence relations on Fn(X,A), which we will call radical congruences (cf. Section 1). We conclude that the lattice of A-algebraic sets in A and the lattice of radical congruences on Fn(X,A) are dually isomorphic. When we choose a general algebra such that Fn(X,A) is the ring of polynomials K[x1, . . . , xn] over an algebraically closed field, we obtain the classical correspondence between algebraic varieties in K and reduced ideals in K[x1, . . . , xn]. In Algebraic Geometry we have a functorial correspondence between algebraic varieties and coordinate algebras K[V ] := K[x1, . . . , xn]/V ⊥. (Here V ⊥ is the ideal of polynomials that vanish on V ). For A-algebraic sets V , we define a coordinate algebra Γ(V ) by Γ(V ) := Fn(X,A)/Φ, where Φ := V ⊥ is the congruence relation corresponding to V . Since A-algebraic sets can be understood as homomorphisms from Fn(X)/Φ to A and since coordinate algebras can be understood as finitely generated subalgebras of a power of A, we get a dual equivalence between the category of A-algebraic sets with polynomial morphisms – yet to be defined – and the category of of finitely generated subalgebras of a power of A with homomorphisms. This result is due to H. Bauer ([Ba83]). In the classical case we get the dual equivalence mentioned afore. We will use the general results to deduce the classical results of Algebraic