Lyapunov decomposition in $$\hbox {d}_{\text {0}}$$ d 0 -algebras
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Lyapunov decomposition in d0 -algebras Anna Avallone1 · Paolo Vitolo1 Received: 14 June 2019 / Accepted: 29 July 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract We prove that a closed d0 -measure on a d0 -algebra can be decomposed into the sum of a Lyapunov d0 -measure and an anti-Lyapunov d0 -measure. Keywords d0 -algebras · BCK-algebras · Measures · Lyapunov decomposition Mathematics Subject Classification 03G25 · 28A12 · 06A12 · 06F35
Introduction In 1974, Kluvanek and Knowles (see [13]) proved a decomposition theorem for a closed σ -additive measure μ on a σ -algebra with values in a quasi-complete locally convex linear space. Precisely, μ can be expressed as the sum of a Lyapunov vector measure and an antiLyapunov vector measure. In this paper we prove a Lyapunov decomposition theorem for closed d0 -measures on d0 -algebras (see Theorem 5.11), which generalizes the decomposition theorem proved in [6]. A similar result has been obtained in [5] for pseudo-d-lattices. We recall that in the 1990s, d-lattices (or, equivalently, lattice-ordered effect algebras) have been introduced as a simultaneous generalization of MV-algebras and orthomodular lattices. Then the investigation of modular measures on d-lattices (see for instance [2–4,6– 8]) gave results which may be applied in both Fuzzy Measure Theory and Non-commutative Measure Theory. Cancellative BCK-algebras (i.e. commutative BCK-algebras with the relative cancellation property, as defined in [10]) generalize MV-algebras, too. As shown in [11], they are related to other fuzzy structures such as clans [17,18,22], Vitali spaces [9] and --semigroups [21]. In 2011, d0 -algebras have been defined, which turned out to be a generalization of both d-lattices and cancellative BCK-algebras (see [14,15]). It is also known that every d0 -algebra is at the same time a ∧-semilattice and a generalized D-poset.
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Paolo Vitolo [email protected] Anna Avallone [email protected]
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Dipartimento di Matematica, Informatica ed Economia, Università della Basilicata, Viale dell’Ateneo Lucano, 10, 85100 Potenza, Italy
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A. Avallone, P. Vitolo
In the paper [16], d0 -measures on d0 -algebras are introduced, of which both modular measures on d-lattices and measures on cancellative BCK-algebras are particular cases. Hence results of the present paper, in addition of generalizing similar ones for modular measures on d-lattices, also lead to Lyapunov decomposition for measures on cancellative BCK-algebras, which, as far as we know, has not been investigated yet. The paper is organized as follows: In the first section we give some preliminaries; in the two subsequent section we collect some facts related to uniformities and measures, respectively. The fourth section is about Lyapunov measures, together with pseudo-non-injective and semiconvex measures, as well as the relationships between them; anti-Lyapunov measure are considered, also. The final section deals with the Lyapunov decomposition, and it ends with the main theorem.
1 Preliminari
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