From a lattice to its ideal lattice

Baker, Kirby A.; Hales, Alfred W.

doi:10.1007/bf02485732

Cited by 27 publications

(18 citation statements)

References 13 publications

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“…In the present context, L(Q(Pfin(co))) is a complete lattice; 1(F) satisfies Whitman's condition as shown by Baker and Hales [2], h: L(Q(Pfin(co))) -» S(Pfin(o))) is a bounded lattice epimorphism as just shown, and 1(F) is isomorphic to a sublattice of S(Pbn((o)) as shown above. It follows that 1(F) is isomorphic to a sublattice of L(Q(P&n(a)))) > and, consequently, 1(F) is isomorphic to a sublattice of L(K), as required.…”

Section: S(pu(o))mentioning

confidence: 74%

𝑄-universal quasivarieties of algebras

Adams¹,

Dziobiak²

1994

Proc. Amer. Math. Soc.

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Abstract. A quasivariety of algebras of finite type is Q-universal if its lattice of subquasivarieties has, as a homomorphic image of a sublattice, the lattice of subquasivarieties of any quasivariety of algebras of finite type. A sufficient condition for a quasivariety to be Q-universal is given, thereby adding, amongst others, the quasivarieties of de Morgan algebras, Kleene algebras, distributive p-algebras, distributive double p-algebras, Heyting algebras, double Heyting algebras, lattices containing the modular lattice A/33 , A/T-algebras, and commutative rings with unity to the known Q-universal quasivarieties.

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Section: S(pu(o))mentioning

confidence: 74%

𝑄-universal quasivarieties of algebras

Adams¹,

Dziobiak²

1994

Proc. Amer. Math. Soc.

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“…A consequence of [20] gives that if P is a semilattice that is the ∧-reduct of a finite distributive lattice, then P is projective in semilattices. We will also need the following slight extension of a result of Baker and Hales [1]. Proof.…”

Section: Theorem 25 [12] a Distributive Lattice Has No Doubly Reducmentioning

confidence: 99%

“…Proof. We follow [1]. Let X be the set of all finite subsets of K partially ordered by set inclusion.…”

Section: Theorem 25 [12] a Distributive Lattice Has No Doubly Reducmentioning

confidence: 99%

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MacNeille transferability and stable classes of Heyting algebras

et al. 2018

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Abstract.A lattice P is transferable for a class of lattices K if whenever P can be embedded into the ideal lattice IK of some K ∈ K, then P can be embedded into K. There is a rich theory of transferability for lattices. Here we introduce the analogous notion of MacNeille transferability, replacing the ideal lattice IK with the MacNeille completion K. Basic properties of MacNeille transferability are developed. Particular attention is paid to MacNeille transferability in the class of Heyting algebras where it relates to stable classes of Heyting algebras, and hence to stable intermediate logics.Mathematics Subject Classification. 06D20, 06B23, 06E15.

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“…B. Nation [4] (iii) follows from (i) and (ii), since a one-to-one map / of a lattice L onto a lattice M is an isomorphism if and only if / and its inverse are order-preserving.…”

Section: Lemma 14 Letv Be Locally Finite Consider the Maps Viamentioning

confidence: 99%

Lattices of pseudovarieties

Aglianò

Nation

1989

J Aust Math Soc A

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We consider the lattice of pseudovarieties contained in a given pseudovariety P. It is shown that if the lattice L of subpseudovarieties of P has finite height, then L is isomorphic to the lattice of subvarieties of a locally finite variety. Thus not every finite lattice is isomorphic to a lattice of subpseudovarieties. Moreover, the lattice of subpseudovarieties of P satisfies every positive universal sentence holding in all the lattices of subvarieties of varieties V(A) generated by algebras A€. P. A pseudovariety P is a class of finite algebras closed under the formation of homomorphic images, subalgebras and finite direct products: in symbols HSPfl n (P) = P. This concept has been useful in many investigations, particularly in the study of various classes of finite semigroups and monoids; see [2], [3], [7], [8] and, for a more general approach, [5].We will consider the lattice of subpseudovarieties of a given pseudovariety P. We will show that several recent results about the lattice of subvarieties of a variety have analogs for pseudovarieties.This investigation originated in a series of discussions between the second author, Kathy Johnston and T. E. Hall at Monash University in August 1986, which produced a direct proof of Corollary 2.6. Hall and Johnston were at tht time working on pseudovarieties of inverse semigroups [8], and Section 5 of that paper contains some interesting results related to the ones herein. C. J. Ash

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From a lattice to its ideal lattice

Cited by 27 publications

References 13 publications

𝑄-universal quasivarieties of algebras

𝑄-universal quasivarieties of algebras

MacNeille transferability and stable classes of Heyting algebras

Lattices of pseudovarieties

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