Generalized quasi-metric semilattices
In [Kōdai Math. Semin. Rep. 22, 443--468 (1970; Zbl 0222.06004)], \textit{Y. Nakamura} introduced entropy for semilattices equipped with a semivaluation in order to provide an abstract approach to the classical measure entropy by Kolmogorov and Sinai. Recently a common approach to various entropies in mathematics (measure entropy, algebraic entropy, topological entropy, set-theoretic entropy, etc.) was proposed by the first two authors [Diss. Math. 542, 1--90 (2019; Zbl 1429.16002)]. This unifying approach does not cover several entropies, starting from the intrinsic algebraic entropy. The definition of the latter entropy, which is closely connected to the more classical algebraic entropy, is based on the fundamental notion of inert subgroup (with respect to an endomorphism), a concept which originally comes from group theory and proved to be very useful in the study of the structure of abelian groups and their endomorphisms. A general approach to cover these ``missing'' intrinsic-like entropies is already elaborated in a forthcoming paper by three of the authors and another two collaborators [\textit{I. Castellano} et al., ``Intrinsic entropy for generalized quasi-metric semilattices'', J. Algebra Appl. (to appear), \url{doi:10.1142/S0219498822502449}]. There the idea is to use a common feature of all these cases, namely, the presence of a semilattice provided with a generalized quasi-metric. The elaboration of generalized quasi-metric semilattices is the aim of the present paper. The paper is very dense in results and for summarizing these, one should (at least) copy-paste the introduction.
scientific article; zbMATH DE number 7481219
scientific article