On Equational Classes of Algebraic Versions of Logic I.

Monk, Donald

doi:10.7146/math.scand.a-10987

Cited by 32 publications

(20 citation statements)

References 3 publications

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“…From this reduction, some properties already proved in [2] easily follow (see Theorem 3.3). We also obtain new results, mainly (1) The determination of the minimum number of variables needed in an equation characterizing a given equational class.…”

Section: Th Lucas1mentioning

confidence: 78%

“…We also obtain new results, mainly (1) The determination of the minimum number of variables needed in an equation characterizing a given equational class. (See Theorem 3.4; this question is raised in [2].) (2) A positive answer for the case a = 1 to the following two questions: Does every equation holding in a CAX also hold in its canonical embedding algebra?…”

Section: Th Lucas1mentioning

confidence: 97%

“…(See Theorem 3.4; this question is raised in [2].) (2) A positive answer for the case a = 1 to the following two questions: Does every equation holding in a CAX also hold in its canonical embedding algebra? Does it also hold in its completion algebra?…”

Section: Th Lucas1mentioning

confidence: 97%

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Equations in the theory of monadic algebras

Lucas¹

1972

Proc. Amer. Math. Soc.

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Identities in the theory of monadic algebras are equivalent to simpler "standard" ones. This property is used to prove some well-known results as well as to determine the minimum number of variables needed in an identity characterizing an equational class. Identities are also shown to be preserved under certain types of extension.In this paper we prove that the equations of the theory of monadic algebras can be "reduced" to simpler "standard" ones. From this reduction, some properties already proved in [2] easily follow (see Theorem 3.3). We also obtain new results, mainly(1) The determination of the minimum number of variables needed in an equation characterizing a given equational class. (See Theorem 3.4; this question is raised in [2].)(2) A positive answer for the case a = 1 to the following two questions: Does every equation holding in a CAX also hold in its canonical embedding algebra? Does it also hold in its completion algebra? (See Corollary 3.6; these questions were raised by Monk at a seminar in algebraic logic at the University of Colorado.) I wish to thank Donald Monk for much valuable advice and Dan Demaree for useful conversations. I am also indebted to the referee whose suggestions have helped to improve a first version of this paper.

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Section: Th Lucas1mentioning

confidence: 78%

Section: Th Lucas1mentioning

confidence: 97%

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Equations in the theory of monadic algebras

Lucas¹

1972

Proc. Amer. Math. Soc.

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“…To put this in perspective, observe that, as shown by Monk [13], the lattice of subvarieties of M is an ω + 1 chain M 0 ⊂ M 1 ⊂ M 2 ⊂ · · · ⊂ M, where M 0 is the trivial variety and M 1 corresponds to the variety of Boolean algebras. Underscoring the difference, it will be seen (from Lemma 3.9) that, for B ∈ M n where n < ω, if B is rigid and non-trivial, then B must be a 2-element algebra.…”

Section: Introductionmentioning

confidence: 99%

Endomorphisms of monadic Boolean algebras

Adams

Dziobiak

2007

Algebra univers.

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A classical result about Boolean algebras independently proved by Magill [10], Maxson [11], and Schein [17] says that non-trivial Boolean algebras are isomorphic whenever their endomorphism monoids are isomorphic. The main point of this note is to show that the finite part of this classical result is true within monadic Boolean algebras. By contrast, there exists a proper class of non-isomorphic (necessarily) infinite monadic Boolean algebras the endomorphism monoid of each of which has only one element (namely, the identity), this being the first known example of a variety that is not universal (in the sense of Hedrlín and Pultr), but contains a proper class of non-isomorphic rigid algebras (that is, the identity is the only endomorphism).

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“…(E4) Monadic Algebras. Let M be the variety of all monadic algebras [14]. By [17, (3) monadic algebra with Boolean reduct 2 n .…”

Section: Applicationsmentioning

confidence: 99%

The Subquasivariety Lattice of a Discriminator Variety

Blanco

Campercholi

Vaggione

2001

Advances in Mathematics

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Let V be a discriminator variety such that the class B=[A # V: A is simple and has no trivial subalgebra] is closed under ultraproducts. This property holds, for example, if V is locally finite or if the language is finite. Let v(V) and q(V) denote the lattice of subvarieties and subquasivarieties of V, respectively. We prove that q(V) is modular iff q(V) is distributive iff v(V) satisfies a certain condition where the case in which the language has a constant symbol is``v(V) is a chain or q(V) =v(V).'' We give an isomorphism between q(V) and a lattice constructed in terms of v(V). Via this isomorphism we characterize the completely meet irreducible (prime) elements of q(V) in terms of the completely meet irreducible elements of v(V). We conclude the paper with applications to the varieties of Boolean algebras, relatively complemented distributive lattices, 4ukasiewicz algebras, Post algebras, complementary semigroups of rank k (x n rx)-rings, R 5 lattices (P-algebras, B-algebras), and monadic algebras. Academic Press

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On Equational Classes of Algebraic Versions of Logic I.

Cited by 32 publications

References 3 publications

Equations in the theory of monadic algebras

Equations in the theory of monadic algebras

Endomorphisms of monadic Boolean algebras

The Subquasivariety Lattice of a Discriminator Variety

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