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A classification of certain group-like FL\(_e\)-chains

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An Erratum to this article was published on 23 June 2015

Abstract

Classification of certain group-like FL\(_e\)-chains is given: We define absorbent-continuity of FL\(_e\)-algebras, along with the notion of subreal chains, and classify absorbent-continuous, group-like FL\(_e\)-algebras over subreal chains: The algebra is determined by its negative cone, and the negative cone can only be chosen from a certain subclass of BL-chains, namely, one with components which are either cancellative (that is, those components are negative cones of totally ordered Abelian groups) or two-element MV-algebras, and with no two consecutive cancellative components. It is shown that the classification theorem does not hold if we drop the absorbent-continuity condition. Our result is the first classification theorem in the literature on FL\(_e\)-algebras that does not assume the condition of being naturally ordered (which, under certain conditions, corresponds to continuity of the monoid operation). In our classification theorem, continuity is replaced by the much weaker absorbent-continuity.

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Notes

  1. Isotonicity of the semigroup operation is not assumed.

  2. He called it reducible.

  3. For residuated integral monoids, divisibility is equivalent to the continuity of the semigroup operation in the order topology if the underlying chain is order dense.

  4. Divisibility is the algebraic analogue of the Intermediate Value Theorem in real analysis, and for residuated integral monoids over order dense chains, it can be considered as a stronger version of continuity of the monoid operation than the continuity of it with respect to the order topology. Indeed, divisibility entails continuity on order dense chains as mentioned in the previous footnote. On the other hand, if the order topology of the chain is discrete then every operation is continuous but obviously not all operations obey the divisibility condition.

  5. The notion of ordinal sum has been slightly modified to ease the formulation of this result.

  6. A group-like FL\(_e\)-algebra is an involutive FL\(_e\)-algebra satisfying \(t=f\); subreal chains are a proper generalization of the chains whose order type is that of \([0,1]\) and will be formally defined later.

  7. Lattice-ordered groups with \({x}\mathbin {\mathbin {\rightarrow _{\circledast }}}{y}:={y} \mathbin {\circledast } {x^{-1}}\) are group-like.

  8. The notation \(\mathbin {\circledast _{co}}\) refers to the fact that if the chain is densely ordered then for a residuated \(\circledast \), its skewed modification (setting \(\bot \) to be the annihilator) is co-residuated, that is, residuated with respect to the dual ordering relation.

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, University of Pécs, Ifjúság u. 6, Pécs, 7624, Hungary

    Sándor Jenei

  2. Department of Knowledge-Based Mathematical Systems, Johannes Kepler University, Altenbergerstrasse 69, Linz, Austria

    Sándor Jenei

  3. Department of Engineering and Mathematical Sciences, University of Siena, San Niccolò, Via Roma 56, 53100 , Siena, Italy

    Franco Montagna

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  1. Sándor Jenei
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  2. Franco Montagna
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Correspondence to Sándor Jenei.

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Sándor Jenei was supported by the OTKA-76811 Grant, the SROP-4.2.1.B-10/2/KONV-2010-0002 Grant, the SROP-4.2.2.C-11/1/KONV-2012-0005 Grant, and the MC ERG Grant 267589.

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Jenei, S., Montagna, F. A classification of certain group-like FL\(_e\)-chains. Synthese 192, 2095–2121 (2015). https://doi.org/10.1007/s11229-014-0409-2

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