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General algebraic geometry and formal concept analysis. (English) Zbl 0940.06006

From the introduction: This paper describes the interaction between classical algebraic geometry, general algebra, and formal concept analysis. 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 \(\mathbb{K}_{n,A}: =(A^n,F_n (X,A)\times F_n(X,A), \perp)\). Here \(A\) is a general algebra, \(F_n(X,A)\) is the free algebra in \(n\) variables in the variety (in the sense of general algebra) \(\text{Var} A\) generated by \(A\), and we have \(\vec a\perp(p,q): \Leftrightarrow p(\vec a)=q (\vec a)\) for \(\vec a\in A^n\) and \(p,q\in F_n(X,A)\). Extents of this formal context will be called \(A\)-algebraic sets. We find that the intents of \(\mathbb{K}\) are certain congruence relations on \(F_n(X,A)\), which we will call radical congruences. We conclude that the lattice of \(A\)-algebraic sets in \(A^n\) and the lattice of radical congruences on \(F_n(X,A)\) are dually isomorphic. When we choose a general algebra such that \(F_n(X,A)\) is the ring of polynomials \(K[x_1, \dots,x_n]\) over an algebraically closed field, we obtain the classical correspondence between algebraic varieties in \(K^n\) and reduced ideals in \(K[x_1, \dots,x_n]\). In algebraic geometry we have a functorial correspondence between algebraic varieties and coordinate algebras \(K[V]:=K[x_1, \dots, x_n]/V^\perp\). (Here \(V^\perp\) is the ideal of polynomials that vanish on \(V)\). For \(A\)-algebraic sets \(V\), we define a coordinate algebra \(\Gamma(V)\) by \(\Gamma(V):= F_n(X,A)/ \Phi\), where \(\Phi:= V^\perp\) is the congruence relation corresponding to \(V\). Since \(A\)-algebraic sets can be understood as homomorphisms from \(F_n(X)/ \Phi\) 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 finitely generated subalgebras of a power of \(A\) with homomorphisms. This result is due to H. Bauer [About Hilbert’s and Rückert’s Nullstellensatz. (FB4-Preprint No. 722, TU Darmstadt) (1983)]. In the classical case we get the dual equivalence mentioned above. We will use the general results to deduce the classical results of algebraic geometry. Along with basic results from formal concept analysis we determine which classical results follow from the general treatment and which results require commutative algebra.

MSC:

06B99 Lattices
14A99 Foundations of algebraic geometry
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06B23 Complete lattices, completions
14A10 Varieties and morphisms
08B20 Free algebras
08A30 Subalgebras, congruence relations
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