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Let La(n, P) be the maximum size of a family of subsets of [n] = {1, 2, … , n} not containing P as a (weak) subposet, and let h(P) be the length of a longest chain in P. The best known upper bound for La(n, P) in terms of |P| and h(P) is due to Chen and Li, who showed that La ( n , P ) ≤ 1 m + 1 | P | + 1 2 ( m 2 + 3 m − 2 ) ( h ( P ) − 1 ) − 1 n ⌊ n / 2 ⌋ ...
L. P. Belluce, A. Di Nola and B. Gerla established a connection between MV-algebras and (dually) lattice ordered semirings by means of so-called coupled semirings. A similar connection was found for basic algebras and semilattice ordered right near semirings by the authors. The aim of this paper is to derive an analogous connection for orthomodular lattices and certain semilattice ordered near semirings...
Set A ⊂ ℕ is less than B ⊂ ℕ in the colex ordering if max(A△B)∈B. In 1980’s, Frankl and Füredi conjectured that the r-uniform graph with m edges consisting of the first m sets of ℕ ( r ) ${\mathbb N}^{(r)}$ in the colex ordering has the largest Lagrangian among all r-uniform graphs with m edges. A result of Motzkin and Straus...
Let F be a non-archimedean linearly ordered field, and C and H be the field of complex numbers and the division algebra of quaternions over F, respectively. In this paper, a class of directed partial orders on C are constructed directly and concretely using additive subgroup of F+. This class of directed partial orders includes those given in Rump and Wang (J. Algebra 400, 1–7, 2014), and Yang (J...
In a pseudocomplemented de Morgan algebra, it is shown that the set of kernel ideals is a complete Heyting lattice, and a necessary and sufficient condition that the set of kernel ideals is boolean (resp. Stone) is derived. In particular, a characterization of a de Morgan Heyting algebra whose congruence lattice is boolean (resp. Stone) is given.
A Birkhoff system is an algebra that has two binary operations ⋅ and + , with each being commutative, associative, and idempotent, and together satisfying x⋅(x + y) = x+(x⋅y). Examples of Birkhoff systems include lattices, and quasilattices, with the latter being the regularization of the variety of lattices. A number of papers have explored the bottom part of the lattice of subvarieties of Birkhoff...
We provide a description of the structure of ℵ0-categorical trees. First the maximal branches of ℵ0-categorical tree are examined, followed by the configuration of the ramification orders, which are then combined to provided necessary and sufficient conditions for a tree to be ℵ0-categorical in terms of these two things.
We show that there is a boolean algebra that has the Freese-Nation property (FN) but not the strong Freese-Nation property (SFN), thus answering a question of Heindorf and Shapiro. Along the way, we produce some new characterizations of the FN and SFN in terms of sequences of elementary submodels.
In this paper we identify and study several lattice structures in the context of directed topology. The set of d-structures on a topological space is a Heyting algebra. The implication is constructed explicitely. There is a Galois connection between the lattice of subsets of the space and the lattice of d-structures which clarifies the idea of removing a subset of the space which has only constant...
An n-ary relation ρ on a set U is strongly rigid if it is preserved only by trivial operations. It is projective if the only idempotent operations in Polρ are projections. Rosenberg, (Rocky Mt. J. Math. 3, 631–639, 1973) characterized all strongly rigid relations on a set with two elements and found a strongly rigid binary relation on every domain U of at least 3 elements. Larose and Tardif (Mult...
This is the second part of a two-part paper on Birkhoff systems. A Birkhoff system is an algebra that has two binary operations ⋅ and + , with each being commutative, associative, and idempotent, and together satisfying x⋅(x + y) = x+(x⋅y). The first part of this paper described the lattice of subvarieties of Birkhoff systems. This second part continues the investigation of subvarieties of Birkhoff...
In this note we consider properties of unital lattice-ordered rings that are division closed and characterize unital lattice-ordered algebras that are algebraic and division closed. Extending partial orders to lattice orders that are division closed is also studied. In particular, it is shown that a field is L∗ if and only if it is O∗.
For a partial order ℙ having infinite antichains by 𝔞 ( ℙ ) $\mathfrak {a}(\mathbb {P})$ we denote the minimal cardinality of an infinite maximal antichain in ℙ and investigate how does this cardinal invariant of posets behave in finite products. In particular we show that min { 𝔞 ( ℙ ) , 𝔭 ( sq ℙ ) } ≤ 𝔞 ( ℙ n ) ≤ 𝔞 ( ℙ ) $\min...
In 1977, Trotter and Moore proved that a poset has dimension at most 3 whenever its cover graph is a forest, or equivalently, has treewidth at most 1. On the other hand, a well-known construction of Kelly shows that there are posets of arbitrarily large dimension whose cover graphs have treewidth 3. In this paper we focus on the boundary case of treewidth 2. It was recently shown that the dimension...
Given a ring and a locally finite poset, an incidence loop or poset loop is obtained from a new and natural extended convolution product on the set of functions mapping intervals of the poset to elements of the ring. The paper investigates the interplay between properties of the ring, the poset, and the loop. The annihilation structure of the ring and extremal elements of the poset determine commutative...
The permutation representation theory of groups has been extended, through quasigroups, to one-sided left (or right) quasigroups. The current paper establishes a link with the theory of ordered sets, introducing the concept of a Burnside order that generalizes the poset of conjugacy classes of subgroups of a finite group. Use of the Burnside order leads to a simplification in the proof of key properties...
Károlyi–Kós and Ardal–Brown–Jungic proved that every vector space over Q has an ordering with no monotone three term arithmetic progression (3-AP). We show that every solvable group has a well ordering with no monotone 6-AP, and each hypoabelian group has an ordering omitting monotone 5-APs. Finally, we prove that every group has a well ordering with no infinite monotone AP.
Every C*-algebra gives rise to an effect module and a convex space of states, which are connected via Kadison duality. We explore this duality in several examples, where the C*-algebra is equipped with the structure of a finite-dimensional Hopf algebra. When the Hopf algebra is the function algebra or group algebra of a finite group, the resulting state spaces form convex monoids. We will prove that...
A subposet Q′ of a poset Q is a copy of a posetP if there is a bijection f between elements of P and Q′ such that x ≤ y in P iff f(x) ≤ f(y) in Q. For posets P, P′, let the poset Ramsey numberR(P, P′) be the smallest N such that no matter how the elements of the Boolean lattice QN are colored red and blue, there is a copy of P with all red elements or a copy of P′ with all blue elements. We provide...
In this article we investigate the lattices of Dyck paths of type A and B under dominance order, and explicitly describe their Heyting algebra structure. This means that each Dyck path of either type has a relative pseudocomplement with respect to some other Dyck path of the same type. While the proof that this lattice forms a Heyting algebra is quite straightforward, the explicit computation of the...
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