L-algebras, normality and exact completions
(2023) XiV Portuguese Category Seminar — Location: Coimbra (13.October.2023)
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- Abstract
- 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.
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Gran, M. (2023). L-algebras, normality and exact completions. Categories and General Algebraic Structures with Applications. Published. XiV Portuguese Category Seminar, Coimbra. https://hdl.handle.net/2078.5/245169 (Original work published 2023)