Mathematica Bohemica

Mathematica Bohemica, first online, pp. 1-23

Multi-argument specialization semilattices

Paolo Lipparini

Received July 13, 2024. Published online January 10, 2025.

Abstract: If $X$ is a closure space with closure $K$, we consider the semilattice $(\mathcal P(X), \cup)$ endowed with a further relation $ x \sqsubseteq\{ y_1, y_2, \dots, y_n\} $ between elements of $\mathcal P(X)$ and finite subsets of $\mathcal P(X)$, whose interpretation is $x \subseteq Ky_1 \cup Ky_2 \cup\dots\cup Ky_n $. We present axioms for such multi-argument specialization semilattices and show that this list of axioms is sound and complete for substructures of closure spaces, namely, a model satisfies the axioms if and only if it can be embedded into the structure associated to a closure space as in the previous sentence. As a main tool for the proof, we provide a canonical embedding of a multi-argument specialization semilattice into (the structure associated to) a closure semilattice.

Keywords: multi-argument specialization semilattice; closure semilattice; closure space; universal extension

Classification MSC: 06A15, 06A12, 54A05, 06F99

DOI: 10.21136/MB.2025.0082-24

PDF available at: Institute of Mathematics CAS

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Affiliations: Paolo Lipparini, Dipartimento di Matematica, Viale della Multa Ricerca Scientifica, Università di Roma "Tor Vergata", I-00133 Rome, Italy, e-mail: lipparin@axp.mat.uniroma2.it

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