Near-unanimity is decomposable

Abstract

In the interpretability lattice, the varieties possessing a near unaninimity term constitute a filter. It is shown that this filter is the proper intersection of two larger filters. One of these two filters is shown to be modular.

Taylor’s modularity conjecture holds for linear idempotent varieties

“…It was shown by Garcia and Taylor [5] that the filter of congruence distributive varieties is the proper intersection of larger filters. A similar result holds for the filter of all varieties with a near-unanimity operation [8].…”

On the congruence modularity conjecture

“…The present author has shown in [16] that a similar result holds for the filter of all varieties with a near-unanimity operation.…”

Between an $n$-ary and an $n{+}1$-ary near-unanimity term

“…Except for the penultimate line, the argument is identical with [30,Theorem 3.19]. Compare also Remark 3.3: from another point of view, the proof of Proposition 5.1 exploits the fact mentioned in 3.3(b) that a variety with an n 1 2 -near-unanimity term has directed Gumm terms; then classical arguments from [11,14,24] can be used in order to get congruence modularity from a sequence of ternary terms.…”

“…(3) When h is an integer the first part follows from [30,Theorem 3.19]. When h is a half-integer it follows from Proposition 5.1.…”

“…More recently, a fundamental paper by Berman, Idziak, Marković, McKenzie, Valeriote, Willard [7] showed that, among congruence distributive varieties, nearunanimity terms play a very important role in tractability problems [15]. Recent results about near-unanimity terms include [4,9,10,27,28,30,31].…”