Bounded homomorphisms and finitely generated fiber products of lattices

William DeMeo; Peter Mayr; Nik Ruskuc

doi:10.1142/S0218196720500174

Bounded homomorphisms and finitely generated fiber products of lattices

William DeMeo, Peter Mayr^*, Nik Ruskuc

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

We investigate when fiber products of lattices are finitely generated and obtain a new characterization of bounded lattice homomorphisms onto lattices satisfying a property we call Dean's condition (D) which arises from Dean's solution to the word problem for finitely presented lattices. In particular, all finitely presented lattices and those satisfying Whitman's condition satisfy (D). For lattice epimorphisms g:A→D, h:B→D, where A, B are finitely generated and D satisfies (D), we show the following: If g and h are bounded, then their fiber product (pullback) C={(a,b)∈A×B | g(a)=h(b)} is finitely generated. While the converse is not true in general, it does hold when A and B are free. As a consequence we obtain an (exponential time) algorithm to decide boundedness for finitely presented lattices and their finitely generated sublattices satisfying (D). This generalizes an unpublished result of Freese and Nation.

Original language	English
Pages (from-to)	693-710
Journal	International Journal of Algebra and Computation
Volume	30
Issue number	04
Early online date	28 Jan 2020
DOIs	https://doi.org/10.1142/S0218196720500174
Publication status	Published - Jun 2020

Keywords

Free lattice
Finitely presented lattice
Whitman's condition
Bounded lattice
Subdirect product
Pullback

Access to Document

10.1142/S0218196720500174

sdl8
Copyright © 2020 World Scientific Publishing Company. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1142/S0218196720500174
Accepted author manuscript, 307 KBLicence: Unspecified

Cite this

@article{4014aff5346b484b9f5b448bf9f1ab65,

title = "Bounded homomorphisms and finitely generated fiber products of lattices",

abstract = "We investigate when fiber products of lattices are finitely generated and obtain a new characterization of bounded lattice homomorphisms onto lattices satisfying a property we call Dean's condition (D) which arises from Dean's solution to the word problem for finitely presented lattices. In particular, all finitely presented lattices and those satisfying Whitman's condition satisfy (D). For lattice epimorphisms g:A→D, h:B→D, where A, B are finitely generated and D satisfies (D), we show the following: If g and h are bounded, then their fiber product (pullback) C={(a,b)∈A×B | g(a)=h(b)} is finitely generated. While the converse is not true in general, it does hold when A and B are free. As a consequence we obtain an (exponential time) algorithm to decide boundedness for finitely presented lattices and their finitely generated sublattices satisfying (D). This generalizes an unpublished result of Freese and Nation.",

keywords = "Free lattice, Finitely presented lattice, Whitman's condition, Bounded lattice, Subdirect product, Pullback",

author = "William DeMeo and Peter Mayr and Nik Ruskuc",

note = "Funding: The first and second authors were supported by the National Science Foundation under Grant No. DMS 1500254.",

year = "2020",

month = jun,

doi = "10.1142/S0218196720500174",

language = "English",

volume = "30",

pages = "693--710",

journal = "International Journal of Algebra and Computation",

issn = "0218-1967",

publisher = "WORLD SCIENTIFIC PUBL CO PTE LTD",

number = "04",

}

TY - JOUR

T1 - Bounded homomorphisms and finitely generated fiber products of lattices

AU - DeMeo, William

AU - Mayr, Peter

AU - Ruskuc, Nik

N1 - Funding: The first and second authors were supported by the National Science Foundation under Grant No. DMS 1500254.

PY - 2020/6

Y1 - 2020/6

N2 - We investigate when fiber products of lattices are finitely generated and obtain a new characterization of bounded lattice homomorphisms onto lattices satisfying a property we call Dean's condition (D) which arises from Dean's solution to the word problem for finitely presented lattices. In particular, all finitely presented lattices and those satisfying Whitman's condition satisfy (D). For lattice epimorphisms g:A→D, h:B→D, where A, B are finitely generated and D satisfies (D), we show the following: If g and h are bounded, then their fiber product (pullback) C={(a,b)∈A×B | g(a)=h(b)} is finitely generated. While the converse is not true in general, it does hold when A and B are free. As a consequence we obtain an (exponential time) algorithm to decide boundedness for finitely presented lattices and their finitely generated sublattices satisfying (D). This generalizes an unpublished result of Freese and Nation.

AB - We investigate when fiber products of lattices are finitely generated and obtain a new characterization of bounded lattice homomorphisms onto lattices satisfying a property we call Dean's condition (D) which arises from Dean's solution to the word problem for finitely presented lattices. In particular, all finitely presented lattices and those satisfying Whitman's condition satisfy (D). For lattice epimorphisms g:A→D, h:B→D, where A, B are finitely generated and D satisfies (D), we show the following: If g and h are bounded, then their fiber product (pullback) C={(a,b)∈A×B | g(a)=h(b)} is finitely generated. While the converse is not true in general, it does hold when A and B are free. As a consequence we obtain an (exponential time) algorithm to decide boundedness for finitely presented lattices and their finitely generated sublattices satisfying (D). This generalizes an unpublished result of Freese and Nation.

KW - Free lattice

KW - Finitely presented lattice

KW - Whitman's condition

KW - Bounded lattice

KW - Subdirect product

KW - Pullback

UR - https://arxiv.org/abs/1907.08046

U2 - 10.1142/S0218196720500174

DO - 10.1142/S0218196720500174

M3 - Article

SN - 0218-1967

VL - 30

SP - 693

EP - 710

JO - International Journal of Algebra and Computation

JF - International Journal of Algebra and Computation

IS - 04

ER -

Bounded homomorphisms and finitely generated fiber products of lattices

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this