The two-sorted algebraic theory of states, and the universal states of MV-algebras

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21230%2F21%3A00350826" target="_blank" >RIV/68407700:21230/21:00350826 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1016/j.jpaa.2021.106771" target="_blank" >https://doi.org/10.1016/j.jpaa.2021.106771</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.jpaa.2021.106771" target="_blank" >10.1016/j.jpaa.2021.106771</a>

Jazyk výsledku

angličtina

Název v původním jazyce

The two-sorted algebraic theory of states, and the universal states of MV-algebras

Popis výsledku v původním jazyce

States of unital Abelian lattice-groups provide an abstraction of expected-value operators. A well-known theorem due to Mundici asserts that the category of unital lattice-groups is equivalent to the algebraic category of MV-algebras, and their homomorphisms. Through this equivalence, states of lattice-groups correspond to certain [0,1]-valued functionals on MV-algebras, which are also known as states. In this paper we allow states to take values in any unital lattice-group (or in any MV-algebra) rather than just in R (or just in [0,1], respectively). We introduce a two-sorted algebraic theory whose models are precisely states of MV-algebras. We extend Mundici's equivalence to one between the category of MV-algebras with states as morphisms, and the category of unital Abelian lattice-groups with, again, states as morphisms. Thus, the models of our two-sorted theory may also be regarded as states between unital Abelian lattice-groups. As our first main result, we derive the existence of the universal state of any MV-algebra from the existence of free algebras in multi-sorted algebraic categories. In the remaining part of the paper, we seek concrete representations of such universal states. We begin by clarifying the relationship of universal states with the theory of affine representations: the universal state A->B of the MV-algebra A coincides with a certain modification of Choquet's affine representation (of the lattice-group corresponding to A) if, and only if, B is semisimple. Locally finite MV-algebras are semisimple, and Boolean algebras are instances of locally finite MV-algebras. Our second main result is then that the universal state of any locally finite MV-algebra has semisimple codomain, and can thus be described through our adaptation of Choquet's affine representation.

Název v anglickém jazyce

The two-sorted algebraic theory of states, and the universal states of MV-algebras

Popis výsledku anglicky

States of unital Abelian lattice-groups provide an abstraction of expected-value operators. A well-known theorem due to Mundici asserts that the category of unital lattice-groups is equivalent to the algebraic category of MV-algebras, and their homomorphisms. Through this equivalence, states of lattice-groups correspond to certain [0,1]-valued functionals on MV-algebras, which are also known as states. In this paper we allow states to take values in any unital lattice-group (or in any MV-algebra) rather than just in R (or just in [0,1], respectively). We introduce a two-sorted algebraic theory whose models are precisely states of MV-algebras. We extend Mundici's equivalence to one between the category of MV-algebras with states as morphisms, and the category of unital Abelian lattice-groups with, again, states as morphisms. Thus, the models of our two-sorted theory may also be regarded as states between unital Abelian lattice-groups. As our first main result, we derive the existence of the universal state of any MV-algebra from the existence of free algebras in multi-sorted algebraic categories. In the remaining part of the paper, we seek concrete representations of such universal states. We begin by clarifying the relationship of universal states with the theory of affine representations: the universal state A->B of the MV-algebra A coincides with a certain modification of Choquet's affine representation (of the lattice-group corresponding to A) if, and only if, B is semisimple. Locally finite MV-algebras are semisimple, and Boolean algebras are instances of locally finite MV-algebras. Our second main result is then that the universal state of any locally finite MV-algebra has semisimple codomain, and can thus be described through our adaptation of Choquet's affine representation.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/EF16_019%2F0000765" target="_blank" >EF16_019/0000765: Výzkumné centrum informatiky</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2021

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Pure and Applied Algebra

ISSN

0022-4049

e-ISSN

1873-1376

Svazek periodika

225

Číslo periodika v rámci svazku

12

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

21

Strana od-do

—

Kód UT WoS článku

000668926000024

EID výsledku v databázi Scopus

2-s2.0-85104705166