Related papers: Order polynomially complete lattices must be LARGE
We investigate the provability of classical combinatorial theorems in ZF. Using combinatorial arguments, we establish the following results for each infinite cardinal ${\kappa}\in On$, (1) ${\kappa}^+\to ({\kappa},{\omega}+1)$, (2) any…
Dualization of a monotone Boolean function on a finite lattice can be represented by transforming the set of its minimal 1 to the set of its maximal 0 values. In this paper we consider finite lattices given by ordered sets of their meet and…
For large ranks, there is no good algorithm that decides whether a given lattice has an orthonormal basis. But when the lattice is given with enough symmetry, we can construct a provably deterministic polynomial-time algorithm to accomplish…
In this note we are concerned with the validity of an uncountable analogue of a combinatorial lemma due to Vlastimil Pt\'ak. We show that the validity of the result for $\omega_1$ can not be decided in ZFC alone. We also provide a…
Suppose L and M are full-rank lattices in Euclidean space, such that vol(L) < vol(M). Answering a question of Han and Wang from 2001, we show how to construct a bounded measurable set F (we can even take F to be a finite union of polytopes)…
Zilber's Theorem states that a finite lattice $L$ is planar if{}f it has a complementary order relation. We provide a new proof for this crucial result and discuss some applications, including a canonical form for finite planar lattices and…
A monotonous polyomino is formed by all lattice unit squares met by the graph of some fixed monotonous continuous function $f:[a,b] \to \mathbb{R}$ with $f(k) \notin \mathbb{Z}$ whenever $k \in \mathbb{Z}$. Our main result says that the…
A set theory is developed based on the approximations of sets and denoted by AS. In AS the set of all sets exists but the argument for Russell's and Cantor's paradox fail. The Axioms of Separation, Replacement and Foundation are not valid.…
We prove that the problems of representing a finite ordered complemented semigroup or finite lattice-ordered semigroup as an algebra of binary relations over a finite set are undecidable. In the case that complementation is taken with…
One of the longstanding problems in universal algebra is the question of which finite lattices are isomorphic to the congruence lattices of finite algebras. This question can be phrased as which finite lattices can be represented as…
Recently it has been proved that, assuming that there is an almost disjoint family of cardinality (2^{\mathfrak c}) in (\mathfrak c) (which is assured, for instance, by either Martin's Axiom, or CH, or even $2^{<\mathfrak c=\mathfrak c$})…
We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V is a model…
We show that the first order theory of the lattice of open sets in some natural topological spaces is $m$-equivalent to second order arithmetic. We also show that for many natural computable metric spaces and computable domains the first…
We prove the following two results. Theorem A: Let alpha be a limit ordinal. Suppose that 2^{|alpha|}<aleph_alpha and 2^{|alpha|^+}<aleph_{|alpha|^+}, whereas aleph_alpha^{|alpha|}>aleph_{|alpha|^+}. Then for all n< omega and for all…
In this paper, we use a simple discrete dynamical model to study integer partitions and their lattice. The set of reachable configurations of the model, with the order induced by the transition rule defined on it, is the lattice of all…
We characterize well-founded algebraic lattices by means of forbidden subsemilattices of the join-semilattice made of their compact elements. More specifically, we show that an algebraic lattice $L$ is well-founded if and only if $K(L)$,…
Higher order set theory has been a topic of interest for some time, with recent efforts focused on the strength of second order set theories [KW16]. In this paper we strive to present one 'theory of collections' that allows for a formal…
We force the Axiom of Choice over the least initial segment of a Nairian model satisfying ZF. In the forcing extension, square_kappa fails at all uncountable cardinals kappa, and every regular cardinal is omega-strongly measurable in HOD,…
Given a bounded lattice $L$ with bounds $0$ and $1$, it is well known that the set $\mathsf{Pol}_{0,1}(L)$ of all $0,1$-preserving polynomials of $L$ forms a natural subclass of the set $\mathsf{C}(L)$ of aggregation functions on $L$. The…
We use forcing over admissible sets to show that, for every ordinal $\alpha$ in a club $C\subset\omega_1$, there are copies of $\alpha$ such that the isomorphism between them is not computable in the join of the complete $\Pi^1_1$ set…