The Riesz Hull of a Semisimple MV-Algebra

AbstractMV-algebras and Riesz MV-algebras are categorically equivalent to abelian lattice-ordered groups with strong unit and, respectively, with Riesz spaces (vector-lattices) with strong unit. A standard construction in the literature of lattice-ordered groups is the vector-lattice hull of an archimedean latticeordered group. Following a similar approach, in this paper we define the Riesz hull of a semisimple MV-algebra.

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Equivalence à la Mundici for commutative lattice-ordered monoids

Algebra Universalis ◽

10.1007/s00012-021-00736-3 ◽

2021 ◽

Vol 82 (3) ◽

Author(s):

Marco Abbadini

Keyword(s):

Unary Operation ◽

Ordered Groups ◽

Lattice Ordered Groups ◽

Mv Algebra ◽

Abelian Lattice ◽

Mv Algebras

AbstractWe provide a generalization of Mundici’s equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category of unital commutative lattice-ordered monoids is equivalent to the category of MV-monoidal algebras. Roughly speaking, unital commutative lattice-ordered monoids are unital Abelian lattice-ordered groups without the unary operation $$x \mapsto -x$$ x ↦ - x . The primitive operations are $$+$$ + , $$\vee $$ ∨ , $$\wedge $$ ∧ , 0, 1, $$-1$$ - 1 . A prime example of these structures is $$\mathbb {R}$$ R , with the obvious interpretation of the operations. Analogously, MV-monoidal algebras are MV-algebras without the negation $$x \mapsto \lnot x$$ x ↦ ¬ x . The primitive operations are $$\oplus $$ ⊕ , $$\odot $$ ⊙ , $$\vee $$ ∨ , $$\wedge $$ ∧ , 0, 1. A motivating example of MV-monoidal algebra is the negation-free reduct of the standard MV-algebra $$[0, 1]\subseteq \mathbb {R}$$ [ 0 , 1 ] ⊆ R . We obtain the original Mundici’s equivalence as a corollary of our main result.

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K-Radical classes and product radical classes of MV-algebras

Mathematica Slovaca ◽

10.2478/s12175-008-0062-7 ◽

2008 ◽

Vol 58 (2) ◽

Cited By ~ 1

Author(s):

Ján Jakubík

Keyword(s):

Analogous Result ◽

Radical Class ◽

Partially Ordered ◽

Ordered Groups ◽

Lattice Ordered Groups ◽

Mv Algebra ◽

One To One ◽

Abelian Lattice ◽

Mv Algebras

AbstractFor an MV-algebra let J 0() be the system of all closed ideals of ; this system is partially ordered by the set-theoretical inclusion. A radical class X of MV-algebras will be called a K-radical class iff, whenever ∈ X and is an MV-algebra with J 0() ≅ J 0(), then ∈ X. An analogous notation for lattice ordered groups was introduced and studied by Conrad. In the present paper we show that there is a one-to-one correspondence between K-radical classes of MV-algebras and K-radical classes of abelian lattice ordered groups. We also prove an analogous result for product radical classes of MV-algebras; product radical classes of lattice ordered groups were studied by Ton.

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Minimal vector lattice covers

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700047286 ◽

1971 ◽

Vol 5 (3) ◽

pp. 331-335 ◽

Cited By ~ 10

Author(s):

Roger D. Bleier

Keyword(s):

Vector Lattice ◽

Lattice Ordered Group ◽

Ordered Group ◽

Vector Lattices ◽

Ordered Groups ◽

Lattice Ordered Groups ◽

Archimedean Lattice

We show that each archimedean lattice-ordered group is contained in a unique (up to isomorphism) minimal archimedean vector lattice. This improves a result of Paul F. Conrad appearing previously in this Bulletin. Moreover, we show that this relationship between archimedean lattice-ordered groups and archimedean vector lattices is functorial.

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A relationship between the category of chain MV-algebras and a subcategory of abelian groups

Mathematica Slovaca ◽

10.1515/ms-2021-0037 ◽

2021 ◽

Vol 71 (4) ◽

pp. 1027-1045

Author(s):

Homeira Pajoohesh

Keyword(s):

Abelian Groups ◽

Ordered Groups ◽

Lattice Ordered Groups ◽

Abelian Lattice ◽

Mv Algebras

Abstract The category of MV-algebras is equivalent to the category of abelian lattice ordered groups with strong units. In this article we introduce the category of circled abelian groups and prove that the category of chain MV-algebras is isomorphic with the category of chain circled abelian groups. In the last section we show that the category of chain MV-algebras is a subcategory of abelian cyclically ordered groups.

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On the Schröder-Bernstein problem for abelian lattice ordered groups and for MV-algebras

Soft Computing ◽

10.1007/s00500-003-0318-7 ◽

2003 ◽

Vol 8 (8) ◽

pp. 581-586 ◽

Cited By ~ 1

Author(s):

J. Jakubík

Keyword(s):

Bernstein Problem ◽

Ordered Groups ◽

Lattice Ordered Groups ◽

Abelian Lattice ◽

Mv Algebras

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Weak relatively uniform convergences on MV-algebras

Mathematica Slovaca ◽

10.2478/s12175-012-0078-x ◽

2013 ◽

Vol 63 (1) ◽

Author(s):

Štefan Černák ◽

Ján Jakubík

Keyword(s):

Present Article ◽

Lattice Ordered Group ◽

Ordered Group ◽

Partially Ordered ◽

Ordered Groups ◽

Lattice Ordered Groups ◽

Mv Algebra ◽

Brouwerian Lattice ◽

Mv Algebras ◽

Natural Way

AbstractWeak relatively uniform convergences (wru-convergences, for short) in lattice ordered groups have been investigated in previous authors’ papers. In the present article, the analogous notion for MV-algebras is studied. The system s(A) of all wru-convergences on an MV-algebra A is considered; this system is partially ordered in a natural way. Assuming that the MV-algebra A is divisible, we prove that s(A) is a Brouwerian lattice and that there exists an isomorphism of s(A) into the system s(G) of all wru-convergences on the lattice ordered group G corresponding to the MV-algebra A. Under the assumption that the MV-algebra A is archimedean and divisible, we investigate atoms and dual atoms in the system s(A).

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