The existence of states based on Glivenko semihoops

Archive for Mathematical Logic 61 (7):1145-1170 (2022)
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Abstract

In this paper, we mainly investigate the existence of states based on the Glivenko theorem in bounded semihoops, which are building blocks for the algebraic semantics for relevant fuzzy logics. First, we extend algebraic formulations of the Glivenko theorem to bounded semihoops and give some characterizations of Glivenko semihoops and regular semihoops. The category of regular semihoops is a reflective subcategory of the category of Glivenko semihoops. Moreover, by means of the negative translation term, we characterize the Glivenko variety. Then we show that the regular semihoop of regular elements of a free algebra in the variety of Glivenko semihoops is free in the corresponding variety of regular semihoops. Similar results are derived for the semihoop of dense elements of free Glivenko semihoops. Finally, we give a purely algebraic method to check the existence of states on Glivenko semihoops. In particular, we prove that a bounded semihoop has Bosbach states if and only if it has a divisible filter, and a bounded semihoop has Riečan states if and only if it has a semi-divisible filter.

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Y Jiang
King's College London

References found in this work

Algebraic foundations of many-valued reasoning.Roberto Cignoli - 1999 - Boston: Kluwer Academic Publishers. Edited by Itala M. L. D'Ottaviano & Daniele Mundici.
States on pseudo MV-Algebras.Anatolij Dvurečenskij - 2001 - Studia Logica 68 (3):301-327.
Subdirectly irreducible state-morphism BL-algebras.Anatolij Dvurečenskij - 2011 - Archive for Mathematical Logic 50 (1-2):145-160.

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