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We describe which pairs 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. ...doi:10.1007/s11083-011-9231-3 fatcat:pqoxep66cfh7hgbsdnqwyrkvbm
We provide sufficient conditions for a 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. ...doi:10.1007/s00010-010-0058-6 fatcat:wifyfgvnfjdvxbrmqlbcq5ohsu
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). ...
We study the algorithmic complexity 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  . ... 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. ...doi:10.1142/s0218196706003116 fatcat:jtcpfbpngfgk5dedrpr4cxyi2e
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. ...
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. ...
Given the congruence 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. ...arXiv:1205.3297v3 fatcat:arfx2r2qgfe7hkzgmdlye7qwpi
Bulletin of the Section of Logic
We consider examples 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. ...doi:10.18778/0138-06220.127.116.11.08 fatcat:74mmkihq6nhyzorbkcw7uxs2oe
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. ...
Given the congruence 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) ...doi:10.1007/s11083-012-9282-0 fatcat:7y32iqunczdpdcscn6y3kn75wy
We discuss initial value problems for time evolution equations in one dimensional space which are expressed by the 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 ...doi:10.14495/jsiaml.14.5 fatcat:d6peyfocpbhvdhbttvqfxemeda
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 ...
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 , Ore  ...
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. ...
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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