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The present work originates in a collection of attempts to solving various open problems, in different topics in mathematics (mainly, but not restricted to, universal algebra and lattice theory), all related by a common feature: How large can the range of a functor be? If it is large on objects, then is it also large on diagrams?
Our main result, CLL (Lemma 3.4.2), involves a construction that turns a diagram $$\vec{A} $$ indexed by a poset P, from a category A, to an object of A, called a condensate $$\vec{A} $$ (cf. Definition 3.1.5). A condensate of $$\vec{A} $$ will be written in the from $$ B\otimes \vec{A} $$ where B is a Boolean algebra with additional structure –we shall say a P-scaled Boolean algebra...
In this chapter we shall finalize the approach to this work’s main result, the Condensate Lifting Lemma (CLL). The statement of CLL involves a “condensate” $$F(X)\otimes \vec{A}$$ . General condensates are defined in Sect. 3.1. The statement of CLL also involves categories A, B, S with functors $$ \Phi :\mathcal{A} \rightarrow \mathcal{S}$$ and $$ \Psi :\mathcal{B} \rightarrow \mathcal{S}$$...
One of the main origins of our work is the first author’s paper Gillibert (Int. J. Algebra Comput. 19(1):1–40, 2009), where it is proved, in particular, that the critical point crit(A;B) between a locally finite variety A and a finitely generated congruence-distributive variety B such that $$ {\rm Con_c}A \nsubseteq {\rm Con_c}B $$ is always less than $$\mathfrak{N}_w $$ . One of the goals...
This chapter is intended to illustrate how to use CLL for solving an open problem of lattice theory, although the statement of that problem does not involve lifting diagrams with respect to functors. We set $$\alpha\beta:=\{(x,z)\in X \times X \mid (Ey\in X)(x,y)\in \alpha {\rm and}(y,z)\in \beta\}$$ , for any binary relations a and ß on a set X, and we say that an algebra A is congruence-permutable...
The assignment that sends a regular ring R to its lattice of all principal right ideals can be naturally extended to a functor, denoted by L (cf. Sect. 1.1.2). An earlier occurrence of a condensate-like construction is provided by the proof in Wehrung (J. Math. Log. 6(1):1–24, 2006, Theo- rem 9.3). This construction turns the non-liftability of a certain 1-lattice endomorphism from Mω (cf...
The discussion undertaken, in Chap. 6, about the functor $$\mathbb{L}$$ on regular rings, can be mimicked for the functor $$\mathbb{V}$$ (nonstable K-theory) introduced in Example 1.1.3, restricted to (von Neumann) regular rings. It is a fundamental open problem in the theory of regular rings whether every conical refinement monoid, of cardinality at most $$\aleph$$ 1, is isomorphic...
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