Prime spectra in modular varieties

Abstract

By using the concept of modular commutator,\ud prime congruences are defined for algebras in modular varieties. Then the\ud prime spectrum of an algebra is defined and various spectral properties are\ud discussed. In particular some conditions are given for the spectrum of an\ud algebra to be homeomorphic to a ring spectrum

Going up and lying over in congruence-modular algebras

“…Since f fulfills GU, it follows that there exists a ψ 1 ∈ Spec(B) such that φ 1 ⊆ ψ 1 and f * (ψ 1 ) = ψ. Since g * (φ 2 ) = φ 1 and g fulfills GU , it follows that there exists a ψ 2 ∈ Spec(C) such that φ 2 ⊆ ψ 2 and g * (ψ 2 ) = ψ 1 . Then…”

“…• any admissible canonical embedding in C fulfills LO. (i) [26,Proposition 1.2,1,v] Cg B (f (α)) = f (α ∨ Ker(f ));…”

“…In order to define GU and LO in this general setting, we need a notion of prime congruence; we have chosen the prime congruences introduced through the notion of commutator, which can be defined in congruence-modular varieties [10, p. 82]. [1] shows that the prime spectra of algebras in semi-degenerate congruence-modular varieties have rich enough properties for developping an interesting mathematical theory concerning GU and LO. While the inverse images of prime ideals through morphisms of commutative rings, bounded distributive lattices, MV-algebras and BL-algebras are again prime ideals, the same does not go for prime congruences in algebras from congruence-modular varieties, in general, and, since this property makes the theory of conditions GU and LO work for these particular kinds of algebras, we have had to restrict our research to morphisms that fulfill this property for prime congruences, which we have called admissible morphisms. However, in many kinds of varieties, all morphisms are admissible; we list such varieties in the final section of this paper.…”

“…(ii) If f fulfills LO and Con(B) is a chain, then f fulfills GU.Proof. (i) Let φ, ψ ∈ Spec(A) and φ 1 ∈ Spec(B) such that f * (φ 1 ) = φ and φ ⊆ ψ.If φ = ψ, then we may take ψ 1 = φ 1 , as in the proof of Lemma 5 1…”

“…From these results we obtain some classes in which all morphisms are admissible and fulfill GU and LO. Throughout this section, A, B shall be members of C and f : A → B shall be a morphism in C.Lemma 8 1…”