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Commuting Polynomial Operations of Distributive Lattices

2011
*
Order
*

We describe which pairs

doi:10.1007/s11083-011-9231-3
fatcat:pqoxep66cfh7hgbsdnqwyrkvbm
*of**distributive**lattice**polynomial**operations**commute*. ... Self-*commuting**lattice**polynomial**operations*. Let L be a*distributive**lattice*, and let f , g be*polynomial**operations**of*L. ... We will now use our theorem on*commuting*pairs*of**distributive**lattice**polynomial**operations*to determine all*commuting*pairs*of**distributive**lattice*term*operations*. ...##
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Self-commuting lattice polynomial functions on chains

2010
*
Aequationes Mathematicae
*

We provide sufficient conditions for a

doi:10.1007/s00010-010-0058-6
fatcat:wifyfgvnfjdvxbrmqlbcq5ohsu
*lattice**polynomial*function to be self-*commuting*. We explicitly describe self-*commuting**polynomial*functions over chains. ... Also, we are grateful to the reviewers for their useful comments which led into improvement*of*the current manuscript. ... We start with the result that provides sufficient conditions for a*polynomial*to be self-*commuting*in the general case*of**distributive**lattices*. Lemma 3.6. Let L be a*distributive**lattice*. ...##
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Page 990 of Mathematical Reviews Vol. 16, Issue 10
[page]

1955
*
Mathematical Reviews
*

The author considers the structure

*of*a system R*of*right invertible, left*distributive**operations*defined on a set M. ... If G is a group, the*operations*Bg(a, b) =ba~'ga, for all g e G, form a system*of*right invertible, left*distributive**operations*denoted by II(G). ...##
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TAYLOR TERMS, CONSTRAINT SATISFACTION AND THE COMPLEXITY OF POLYNOMIAL EQUATIONS OVER FINITE ALGEBRAS

2006
*
International journal of algebra and computation
*

We study the algorithmic complexity

doi:10.1142/s0218196706003116
fatcat:jtcpfbpngfgk5dedrpr4cxyi2e
*of*determining whether a system*of**polynomial*equations over a finite algebra admits a solution. ... In particular, we prove a dichotomy result which encompasses the cases*of**lattices*, rings, modules, quasigroups and also generalizes a result*of*Goldmann and Russell for groups [15] . ... We thank Michal Koucký for his nice reduction in Theorem 2.2 (2) and Matt Valeriote for the simplifying argument*of*Lemma 3.10. The first author's research is supported by a grant from NSERC. ...##
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Page 1727 of Mathematical Reviews Vol. , Issue 84e
[page]

1984
*
Mathematical Reviews
*

A

*lattice*L with minimum element 0 is weakly modular [weakly*distributive*] if it satisfies the usual modular law [*distributivity*law*of*join over meet] for x, y and z such that y/\z #0. ... A bisemilattice is a set with two binary*operations*+ and -, each*of*which is idempotent,*commutative*and associative. ...##
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Page 574 of Mathematical Reviews Vol. 17, Issue 6
[page]

1956
*
Mathematical Reviews
*

For this reason the author is led to study the sublattice

*of*those closure*operators*which*commute*with all closure*operators*. This he calls the center*of*the*lattice**of*closure*operators*. ... Center*of*closure*operators*and a decomposition*of*a*lattice*. Math. Japon. 3 (1954), 49-52. This deals with the*lattice**of*all closure*operators*on a*lattice*. ...##
###
Sequences of commutator operations
[article]

2012
*
arXiv
*
pre-print

Given the congruence

arXiv:1205.3297v3
fatcat:arfx2r2qgfe7hkzgmdlye7qwpi
*lattice*L*of*a finite algebra A with a Mal'cev term, we look for those sequences*of**operations*on L that are sequences*of*higher*commutator**operations**of*expansions*of*A. ... The properties*of*higher*commutators*proved so far delimit the number*of*such sequences: the number is always at most countably infinite; if it is infinite, then L is the union*of*two proper subintervals ... As for the binary term condition*commutator*, these*commutator**operations*are completely determined by the clone*of**polynomial*functions*of*an algebra. ...##
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Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras

2016
*
Bulletin of the Section of Logic
*

We consider examples

doi:10.18778/0138-0680.45.3.4.08
fatcat:74mmkihq6nhyzorbkcw7uxs2oe
*of*frontal Heyting algebras, in particular Heyting algebras with the successor, γ and G*operations*as well as expansions*of*some*commutative*integral residuated*lattices*with successor ... We show that adding compatible*operations*to Heyting algebras and to*commutative*residuated*lattices*, both satisfying the Stone law ¬x ∨ ¬¬x = 1, preserves filtering (or directed) unification, that is, ... Hence we can extend the result by allowing expansions*of*Heyting algebras and*of**commutative*integral (even non-*distributive*) residuated*lattices*with compatible*operations*. ...##
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Page 270 of Duke Mathematical Journal Vol. 5, Issue 2
[page]

1939
*
Duke Mathematical Journal
*

However, the conditions given by Ward are more stringent than those satisfied by actual instances

*of*non-*commutative*arith- metic, for example, quotient*lattices*and non-*commutative**polynomial*theory ( ... With these definitions = is a*lattice*which is modular or*distributive*if and only if L is modular or*distributive*. ...##
###
Sequences of Commutator Operations

2013
*
Order
*

Given the congruence

doi:10.1007/s11083-012-9282-0
fatcat:7y32iqunczdpdcscn6y3kn75wy
*lattice*L*of*a finite algebra A with a Mal'cev term, we look for those sequences*of**operations*on L that are sequences*of*higher*commutator**operations**of*expansions*of*A. ... The properties*of*higher*commutators*proved so far delimit the number*of*such sequences: the number is always at most countably infinite; if it is infinite, then L is the union*of*two proper subintervals ... Open Access This article is*distributed*under the terms*of*the Creative Commons Attribution License which permits any use,*distribution*, and reproduction in any medium, provided the original author(s) ...##
###
Lattice equations and their solutions with complexity of polynomial class

2022
*
JSIAM Letters
*

We discuss initial value problems for time evolution equations in one dimensional space which are expressed by the

doi:10.14495/jsiaml.14.5
fatcat:d6peyfocpbhvdhbttvqfxemeda
*lattice**operators*and propose some new equations to which complexity*of*solutions is*of*...*polynomial*class. ... Ikegami et al. reported*lattice*equations*of**polynomial*class which are generated by simple combination*of**lattice**operators*and related them to elementary cellular automata (ECA) as the special case*of*...##
###
Page 5054 of Mathematical Reviews Vol. , Issue 82m
[page]

1982
*
Mathematical Reviews
*

A [modular,

*distributive*] p-algebra is an algebra (L; V,/\,*,0,1) in which the deletion*of*the unary*operation** yields a bounded [modular,*distributive*]*lattice*and in which * satisfies x<a* if and only ... Among the further results*of*this paper are: (1) Every finite*lattice*can be embedded into a*polynomially*complete polarity*lattice*. (2) A complete geometric polarity*lattice*P is*polynomially*complete ...##
###
Page 271 of Duke Mathematical Journal Vol. 5, Issue 2
[page]

1939
*
Duke Mathematical Journal
*

When we have a

*commutative*multiplication, the connection*of*the multi- plication with the*lattice**operations*automatically makes the*lattice*modular (in fact,*distributive*(Ward-Dilworth [1, 2])). ... ‘That a non-*commutative**polynomial*domain is modular can be easily seen from the fact that the degree*of*a*polynomial*is a rank function over the*lattice*in the sense*of*Birkhoff (Birkhoff [1], Ore [2] ...##
###
Page 3787 of Mathematical Reviews Vol. , Issue 80J
[page]

1980
*
Mathematical Reviews
*

For groups this

*operation*is the ordinary*commutator**of*normal subgroups. For J and J ideals in a ring, [/,J]= JJ +JI. ... Let ‘V be a variety*of*algebras with modular congruence*lattices*. The authors define a binary*operation*[a, 8], called the commu- tator*of*a and B, on the congruence*lattices**of*the members*of*‘V. ...##
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Page 6116 of Mathematical Reviews Vol. , Issue 87k
[page]

1987
*
Mathematical Reviews
*

A

*polynomial*f € P,, is said to be meet-Frattini if for every p,q € P,, orthomodular*lattice*Z and a;,---,a, € L, the element p(a;,---,@,)*commutes*with g(a;,---,@n)Af(a1,---,@n) if and only if p(a,,-- ... classes*of*elements*of*the free*distributive**lattice*D7 (under a certain equiv- alence relation) equals 490013 148. ...
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