Some universal properties of the category of clones

Abstract

In [6], O. C. García and W. Taylor asked if the breadth of the lattice of interpretability types of varieties is uncountable. The present paper solves the problem by two different constructions. Both of them show that any cardinal number is the cardinality of an antichain in the named lattice and that the existence of a proper class antichain is equivalent to the negation of Vopěnka's principle. The first construction gives in a way a minimal solution of the problem, whereas the second one gives stronger results about the category of clones. IntroductionThe lattice L of interpretability types of varieties of (finitary monosorted) universal algebras was introduced and started to be investigated in [14]. Then an issue [6] of Memoirs of the AMS by O. C. García and W. Taylor was devoted to the study of L. Many open problems are formulated at the end of [6]. Here, we solve one of them, namely the question what are the possible cardinalities of antichains in L (see [6], page 118, Problem 5; in [6], only countable antichains have been constructed). We prove that (1) every cardinal number can be the cardinality of an antichain in L and (2) the existence of a proper-class antichain in L is equivalent to the negation of the set-theoretical Vopěnka's principle.Vopěnka's principle is examined e. g. in monographs [9] and [1], where some further references are given. We present its formulation and its connection to some large cardinals in Section 1.Presented by A. Szendrei.

Mal’cev Conditions Revisited

“…Proof It follows from Theorem 2.4 and the fact that, assuming the negation of Vopěnka's principle, the large discrete category can be fully embedded into Th (see [25], (3) after Theorem 1.1).…”

Problem list from Algebras, Lattices and Varieties: A conference in honor of Walter Taylor, University of Colorado, 15–18 August, 2004

“…In 2002, V. Trnková and A. Barkhudaryan proved [48] that for every cardinal κ there is an antichain of power κ in L . They also proved that the existence of a proper-class antichain is equivalent to the negation of Vopěnka's principle (a proposed higher axiom of set theory).…”

The category of varieties and interpretations is alg-universal

“…Then an issue [2] of Memoirs of the AMS was devoted to the study of L. One of the many open problems formulated there, whether the breadth of this lattice is uncountable, was solved in [3]. The authors proved there (among other) that every poset can be embedded into L and that the existence of a proper class antichain is equivalent to the negation of Vopěnka's principle (see [4]).…”

“…Moreover, our construction uses idempotent clones only, while the constructions in [3] use many constant operations. To state the main result, let us use an alternative formulation (see the next paragraph):…”