Involution-rigidness — a new exactness property, and its weak version

Abstract

In this paper, we introduce and study a new exactness property in the sense of categorical algebra, which can be seen as a natural strengthening of the well-known Mal’tsev property. A variety of universal algebras has this exactness property if and only if its algebraic theory contains binary terms [Formula: see text] and an [Formula: see text]-ary term [Formula: see text] satisfying [Formula: see text], and [Formula: see text] for each [Formula: see text]. We also study the “weak” version of this exactness property, which similarly strengthens the “weak Mal’tsev property” in the sense of the second author.