Erratum to: On the coset structure of a skew lattice

Abstract

The following is an errata to the paper “On the coset structure of a skew lattice” published in Demonstratio Mathematica as [4]. All the corrections are relevant to the clear understanding of the paper by the reader. The author thanks to K. Cvetko-Vah and J. Leech for discussions that lead to these corrections. This survey paper revisits some of the known decompositions in the theory of skew lattices (non-commutative generalizations of lattices) including both Leech’s decomposition theorems, and some of the base knowledge regarding the coset decomposition. In the introduction it states that skew lattices “consist(s) of the left version of Slavik’s algebras” which, in fact, should state that “Slavik’s algebras lead to a left version of Leech’s skew lattices”. This statement is referred by J. Leech in his review paper [2] and it is proved in Proposition 6 of the present paper, followed by the corresponding discussion. At a late stage of preparation of the paper an error occured into the statement of Proposition 5 on p. 678. The Proposition is assigned to the preprint [3] by J. Leech and M. Kinyon (which remains today to be a preprint, soon to be submitted according to the authors). However, a confusion arises due to the definition of ∧-distributive and ∨distributive skew lattices. By the way they are defined in this paper (using axioms S21, S22 and S23, S24, respectively notice the existing typo) Proposition 5 is not as in the cited reference, and moreover it fails to be true. Therefore, Proposition 5 as stated in the paper is neither a citation of their result nor true. To overcome this confusion, the identities S21 and S22 that define ∧-distributive skew lattices in this paper should be replaced by the identity: x ∧ (y ∨ z) ∧ x ≈ (x ∧ y ∧ x) ∨ (x ∧ z ∧ x)