L-algebras, normality and exact completions

We relate the recent theory of L-algebras with some exactness properties and notions in categorical algebra. We observe that the category of L-algebras is subtractive and normal in the sense of Zurab Janelidze, but neither the category of L-algebras nor that of pre-L-algebras (also called unital cycloids in the literature) are Mal'tsev categories. The commutator of two ideals on an L-algebra is shown to coincide with their intersection. The variety of pre-L-algebras has been recently shown to be the exact completion of the quasi-variety of L-algebras 1. This is a new example of a situation where the exact completion of a normal category is not normal (first observed in 2). Under this respect, the normality property behaves quite differently from most of the classical exactness properties in categorical algebra, such as being subtractive, Mal'tsev or protomodular 3. The talk is based on a joint work with Alberto Facchini and Mara Pompili 4. References: 1 W. Rump, The category of L-algebras, Theory Appl. Categories 39 (2023) 598-624. 2 M. Gran and Z. Janelidze, Star-regularity and regular completions, J. Pure Appl. Algebra 218 (2014) 1771-1782. 3 M. Gran and S. Lack, Semi-localizations of semi-abelian categories, J. Algebra 454 (2016) 206-232. 4 A. Facchini, M. Gran and M. Pompili, Ideals and congruences in L-algebras and pre-L-algebras, arXiv:2305.19042, May 2023.