Generalized quasi-metric semilattices
type: Item
instance of: scholarly article
title: Generalized quasi-metric semilattices
zbMATH Open document ID: 1490.06003
author: Dikran Dikranjan
author: Anna Giordano Bruno
author: Nicolò Zava
author: D. Toller
author: Hans Peter Künzi
published in: Topology and its Applications
publication date: 2022-02-25T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 06A12
Mathematics Subject Classification ID: 54E15
Mathematics Subject Classification ID: 54E35
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 16B50
Mathematics Subject Classification ID: 22D40
Mathematics Subject Classification ID: 37A35
zbMATH DE Number: 7481219
zbMATH Keywords: generalized quasi-metric semilattices
zbMATH Keywords: partial orders
zbMATH Keywords: semilattices
zbMATH Keywords: ascending path condition
zbMATH Keywords: semivaluations
zbMATH Keywords: products
zbMATH Keywords: intrinsic semilattice entropy
zbMATH Keywords: semigroup entropy
reviewed by: Grigore Călugăreanu
MaRDI profile type: Publication
full work available at URL: j.topol.2021.107916
OpenAlex ID: W3212017972
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DOI: 10.1016/J.TOPOL.2021.107916
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