Lattices do not distribute over powerset

Julián Salamanca Téllez

doi:10.1007/s00012-020-00680-8

Lattices do not distribute over powerset

Algebra Universalis ◽

10.1007/s00012-020-00680-8 ◽

2020 ◽

Vol 81 (4) ◽

Author(s):

Julián Salamanca Téllez

Keyword(s):

Free Lattice ◽

Modular Lattices ◽

Distributive Law

AbstractWe show that there is no distributive law of the free lattice monad over the powerset monad. The proof presented here also works for other classes of lattices such as (bounded) distributive/modular lattices and also for some variants of the powerset monad such as the (nonempty) finite powerset monad.

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Families of quasigroup operations satisfying the generalized distributive law

Discrete Mathematics and Applications ◽

10.1515/dma-2020-0018 ◽

2020 ◽

Vol 30 (3) ◽

pp. 187-202

Author(s):

Sergey V. Polin

Keyword(s):

System Of Equations ◽

Systems Of Equations ◽

The Family ◽

Distributive Law ◽

Elimination Of Variables

AbstractThe previous paper was concerned with systems of equations over a certain family 𝓢 of quasigroups. In that work a method of elimination of an outermost variable from the system of equations was suggested and it was shown that further elimination of variables requires that the family 𝓢 of quasigroups satisfy the generalized distributive law (GDL). In this paper we describe families 𝓢 that satisfy GDL. The results are applied to construct classes of easily solvable systems of equations.

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Rough Approximation Operators on a Complete Orthomodular Lattice

Axioms ◽

10.3390/axioms10030164 ◽

2021 ◽

Vol 10 (3) ◽

pp. 164

Author(s):

Songsong Dai

Keyword(s):

Boolean Algebra ◽

Orthomodular Lattice ◽

Orthomodular Lattices ◽

Approximation Operators ◽

Rough Approximation ◽

Distributive Law ◽

Complete Orthomodular Lattice ◽

The Relationship

This paper studies rough approximation via join and meet on a complete orthomodular lattice. Different from Boolean algebra, the distributive law of join over meet does not hold in orthomodular lattices. Some properties of rough approximation rely on the distributive law. Furthermore, we study the relationship among the distributive law, rough approximation and orthomodular lattice-valued relation.

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Seven-modular lattices and a septic base Jacobi identity

Journal of Number Theory ◽

10.1016/s0022-314x(02)00069-0 ◽

2003 ◽

Vol 99 (2) ◽

pp. 361-372 ◽

Cited By ~ 5

Author(s):

Heng Huat Chan ◽

Kok Seng Chua ◽

Patrick Solé

Keyword(s):

Jacobi Identity ◽

Modular Lattices

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Matchings and Radon transforms in lattices II. Concordant sets

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100066573 ◽

1987 ◽

Vol 101 (2) ◽

pp. 221-231 ◽

Cited By ~ 6

Author(s):

Joseph P. S. Kung

Keyword(s):

Incidence Matrix ◽

Finite Lattice ◽

Möbius Function ◽

Radon Transforms ◽

Modular Lattices ◽

Covering Theorem ◽

Mobius Function ◽

Finite Lattices ◽

Semimodular Lattices

AbstractLet and ℳ be subsets of a finite lattice L. is said to be concordant with ℳ if, for every element x in L, either x is in ℳ or there exists an element x+ such that (CS1) the Möbius function μ(x, x+) ≠ 0 and (CS2) for every element j in , x ∨ j ≠ x+. We prove that if is concordant with ℳ, then the incidence matrix I(ℳ | ) has maximum possible rank ||, and hence there exists an injection σ: → ℳ such that σ(j) ≥ j for all j in . Using this, we derive several rank and covering inequalities in finite lattices. Among the results are generalizations of the Dowling-Wilson inequalities and Dilworth's covering theorem to semimodular lattices, and a refinement of Dilworth's covering theorem for modular lattices.

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