The tag has no wiki summary.

learn more… | top users | synonyms

3
votes
1answer
133 views

Profinite completion of a partial order

In Johnstone's Stone Spaces it is proved that the category of profinite partial orders is (equivalent to) the category of ordered Stone spaces (also called Priestley spaces) and that the obvious ...
6
votes
2answers
570 views

Given a cardinal k, what's the biggest dense linear order with a dense subset of size k?

It's not hard to show that for any cardinal $\kappa$, there is no dense linear order without endpoints (DLO) of size greater than $2^{\kappa}$ that has a dense subset of size $\kappa$. But one can ...
2
votes
0answers
152 views

An equivariant Hahn Embedding Theorem?

The Hahn Embedding Theorem asserts that for any (linearly) ordered abelian group $\Lambda$, there exists a linearly ordered indexing set $\Omega$ such that $\Lambda$ admits an order-preserving group ...
4
votes
2answers
217 views

Equality-preserving embeddings of finite trees

For finite trees $T_{1}$ and $T_{2}$ labelled by elements of some infinite set $S$, (we may assume that $S=\mathbb{N}$ without loss of generality), we define an equality-preserving embedding $f$ to be ...
3
votes
2answers
141 views

Finite lattices whose number of join-irreducibles does not exceed its height

In a finite distributive lattice $L$ one has $height(L) = |J(L)|$ i.e. the size of the largest chain equals the number of join-irreducible elements. Briefly, this follows by arranging the subposet ...
1
vote
1answer
343 views

Compactness and completeness in Gödel logic

The standard proof of the completeness theorem in first-order Gödel logic is based on a first-order countable language. I want to know that is there any proof of the completeness theorem in ...
7
votes
1answer
172 views

Status of Barany's conjecture?

One of Barany's most intriguing conjectures is about the $f$-vectors of convex polytopes. It asks: Let $P$ be a convex $d$-polytope. Is it always true that $f_k \geq \min(f_0, f_{d-1})$? A ...
0
votes
1answer
187 views

Examples of groups such that order isomorphism of the subgroups of $G\times G$ and $H\times H$ does not imply isomorphism of $G$ and $H$

Let $G$ and $H$ be groups, $\operatorname{Sub}(G\times G)$ be the set of all subgroups of $G\times G$ and $\operatorname{Sub}(H\times H)$ be the set of all subgroups of $H\times H$. Assume there ...
3
votes
2answers
166 views

Combinatorial design for minimization problem over binary strings

Suppose the cost of a binary string $B$ of length $k$ is the number of $1$s that occur before the last $0$. For example, $1110$ has cost 3 while $0111$ has cost 0. Now suppose you can choose $k$ ...
10
votes
1answer
533 views

Can all aleph_2-dense subsets of R be isomorphic?

Let $\kappa$ be an infinite cardinal. For a subset $A \subseteq \mathbb{R}$, we say that $A$ is $\kappa$-dense if $|A \cap (a, b)| = \kappa$ for every interval $(a, b)$. By Cantor, any two ...
4
votes
2answers
153 views

Embedding a linearly ordered free monoid into a linearly ordered group

A linearly ordered (shortly, l.o.) monoid is a triple $\mathbb M = (M, \cdot, \le)$ for which $(M, \cdot)$ is a (multiplicatively written) monoid and $\le$ is a total order on $M$ such that $xy < ...
1
vote
1answer
100 views

Linearly ordered set arithmetic: reference request

A lot has been written about the arithmetic of ordinal numbers. However, we can also do arithmetic with linearly ordered sets. Question. Is there an article or book where I can learn the basics of ...
14
votes
3answers
807 views

Subposets of small Dushnik-Miller dimension

The Dushnik–Miller dimension of a partial order $(P,{\leq})$ is the smallest possible size $d$ for a family ${\leq_1},\ldots,{\leq_d}$ of total orderings of $P$ whose intersection is ${\leq}$, ...
7
votes
2answers
308 views

Which linearly ordered sets have the property that their completion is equipotent with their powerset?

As is well-known, ZFC proves the equipotency of $\mathbb{R}$ and $\mathcal{P}(\mathbb{Q}).$ Is there a nice characterization of those linearly ordered sets $L$ which, like $\mathbb{Q}$, have the ...
2
votes
1answer
87 views

Has the single sorted case of formal concept analysis been investigated?

A formal context in formal concept analysis is a triple $K = (G, M, I)$ where $G$ is a set of objects, $M$ is a set of attributes and the binary relation $I \subset G \times M$ shows which objects ...
-2
votes
1answer
77 views

Monotonic sequence (edited) [closed]

For any two n-dim vector $v$ and $v'$ define $v\leq v'$ iff for each $1\leq i\leq n$, $v_i\leq v_i'$. Suppose further that the entry of vectors can only take values from $m$ distinct values $\{a_1, ...
3
votes
1answer
72 views

existence of more distributive Boolean lattices

Are there a Boolean lattice $(X,\le)$ and a nonempty collection $(a_{ij})_{i\in I,j\in J}\subseteq X$ such that $$\bigvee_{i\in I}\bigwedge_{j\in J}{a_{ij}}\ne \bigwedge_{f:I\to J}\bigvee_{i\in ...
0
votes
1answer
91 views

Characterizing Inf and Sup sets

Suppose $X$ is a set. Let $\Xi$ be the set of all pairs $(\mathcal A,\mathcal B)$ such that $$\mathcal A, \mathcal B \subseteq \mathcal P(X)$$ and for some partial order $\le$ on $X$, $\mathcal A$ is ...
4
votes
0answers
64 views

Reference for statement that almost every $n$-element partial order has trivial automorphism group

I'm looking for a reference for the statement that almost every partial order on $n$ elements has trivial automorphism group. I've been told that this is a folklore result. Does anyone know of a ...
1
vote
0answers
57 views

First-order Peano Axioms and order-completeness of $\mathbb{N}$ [closed]

Definition: An ordered set is order-complete if any nonempty subset with an upper bound, has a lowest upper bound or supremo. Notation: We denote the system of first-order Peano Axioms (along with ...
8
votes
0answers
257 views

Higher-order dimension in posets: a reference request

Let $P = (X, \le)$ be a partially-ordered set. Then the dimension of $P$ is the minimum number of total orders over $X$ whose intersection yields $P$. Alternately, the dimension of $P$ is the minimum ...
4
votes
2answers
294 views

Projections in a W*-algebra as a continuous lattice?

A continuous lattice is a complete lattice $L$ in which every element $y$ is equal to $\bigvee${$x \in L \mid x \ll y$} where $x \ll y$ ("x approximates y" or "x is way below y") if for any directed ...
2
votes
1answer
123 views

Semitransitive relations

By a digraph, let us mean an ordered pair $(X,r)$ with $r : X \times X \rightarrow B,$ where $X$ is a set and $B = \{\mathrm{False}, \mathrm{True}\}.$ Then supposing $\mathbb{X} =(X,r)$ is a digraph, ...
2
votes
0answers
54 views

A categorical analogue of Debreu's independent factors theorem

Background A major question in Decision Theory is that of the cardinal meaning of a utility function. That is, given a set $X$, a utility function $u:X\rightarrow \mathbb{R}$ represents the choices ...
1
vote
0answers
193 views

What is known about orbifolding ordered groups and sets? Who has been involved? Links to Lee metrics?

In mathematical music theory several ordered groups are considered. Some examples contain the frequency space or Tonnetzes. Other groups (commutative and non-commutative ones) are discussed by Dawid ...
19
votes
4answers
829 views

When does a Galois connection induce a topology?

Let $(X,\leq)$ and $(Y,\leq)$ by partially ordered sets. Recall that a(n antitone) Galois connection between $X$ and $Y$ is a pair of order-reversing maps $\Phi: X \rightarrow Y, \ \Psi: Y ...
5
votes
0answers
162 views

Proving equivalence of a tree-based version of Countable Choice for families of finite sets

In this paper by Good and Tree, the following result is mentioned without proof as part of Proposition 6.5: Each of the following statements imply those beneath it. The countable union of ...
2
votes
3answers
117 views

name for a subset of a binary relational structure which is “closed downward”?

Let $X$ be a set and suppose that $R$ is a binary relation on $X$. Suppose further that $S\subseteq X$ has the property that whenever $y\in S$ and $xRy$ ($x\in X$), then also $x\in S$. Does such an ...
1
vote
1answer
70 views

Uniqueness of minimal completions of a partially ordered set

The partially ordered set $(Y,\le)$ is called a superspace of the partially ordered set $(X,\le')$ iff $X\subseteq Y$. $\le'=(\le\cap X^2)$. and a completion of $(X,\le')$ if in addition $~~ 3$. ...
16
votes
2answers
717 views

An order type $\tau$ equal to its power $\tau^n, n>2$

(This is a re-post of my old unanswered question from Math.SE) For purposes of this question, let's concern ourselves only with linear (but not necessarily well-founded) order types. Recall that: ...
2
votes
1answer
87 views

Example involving partially ordered Abelian groups

Definition 1: Let $(G,\leq)$ be a nonzero partially ordered Abelian group with order unit $u$. (Recall that $u\in G$ is a order unit if, for every $g\in G$, there exists $N\in\mathbb N$ such that ...
2
votes
1answer
201 views

Is any order bounded continuous linear functionals a difference of positive continuous functionals?

Let $B$ be a Banach space and $K$ a closed proper cone in $B$ such that the induced partial order makes $B$ a vector lattice. Let $K'=\{x^*\in B':\langle x^*, x\rangle\geq 0\ \forall x\in K\}$ the ...
4
votes
1answer
122 views

Totally right preorderable groups

Are there any known non-trivial sufficient conditions, or full characterizations, of a totally right-preorderable group? More precisely: totally right-preorderable: has a non-trivial total ...
2
votes
2answers
279 views

Embedding a brouwerian lattice into a boolean lattice

I have already asked a similar question at http://math.stackexchange.com/questions/470704/can-a-brouwerian-lattice-be-extended-into-a-boolean-algebra but have received no answer. Sorry, I ask a ...
1
vote
0answers
298 views

Functoriality of a standard integral domain construction.

The evident forgetful functor from fields to integral domains has a left adjoint, namely the construction of the quotient field for a given integral domain. Another standard construction is taking the ...
3
votes
0answers
86 views

Expected size of $k$-th layer of a POSET

Is this known? What is the expected width of the $k$-th layer (anti-chain layer) of a $d$-dimensional partially ordered set of $n$ elements formed by product of $d$ random linear orders chosen from ...
9
votes
2answers
281 views

Extending a partial order while preserving an automorphism

It is well known that if $(P, \leq)$ is a partial order then $\leq$ can always be extended to a linear order. This is sometimes called Szpilrajn´s theorem although it had been previously proved by ...
3
votes
3answers
217 views

Name for “lower/upper bounds” of arbitrary relations?

Given a partial order $R_{\leq}$ over a set $D$, the set of upper bounds under $R$ of a subset $S$ of $D$ is commonly defined as $\{ y \in D | \ \forall x\in S, x R y \}$. (The set of lower bounds of ...
5
votes
1answer
253 views

Rotation-invariant strict-inclusion-preserving preorderings on subsets of the circle

Say that a preordering $\le$ on a set of subsets of some space preserves strict inclusion provided that $A\lt B$ whenever $A\subset B$ (where $A\lt B$ iff $A\le B$ and $B \not\le A$). Let the space ...
2
votes
1answer
111 views

Counting a type of self-dual poset

An exercise in Stanley's Enumerative Combinatorics (Chapter 3, ex. 8) asked for an example of a finite self-dual poset, (i.e. there is a bijection $f: P\to P$ such that $s\le t \Longleftrightarrow ...
4
votes
1answer
173 views

Does every locally finite acyclic directed set embed into a linear order locally isomorphic to the integers? (Edit: extend, not merely embed.)

Let $S=(S,\prec)$ be a set together with an acyclic binary relation, generally nontransitive. $S$ is locally finite if, for every element $x\in S$, the sets $\{w|w\prec x\}$ ("direct past of $x$") ...
8
votes
1answer
204 views

Does every countably infinite interval-finite partial order embed into the integers?

A partially ordered set $(S,\le)$ is called interval finite if the open intervals $(x,z):=\{y|x\le y\le z\}$ are finite for all choices of $x,z$ in $S$. An embedding $(S,\le)\rightarrow(S',\le')$ of ...
2
votes
1answer
129 views

Linear order extensions on (nonabelian) groups

If $G$ is a group with a (left) linear order, does every (left) partial order on $G$ extend to a (left) linear order? The answer is affirmative on abelian groups, where being torsion-free is ...
2
votes
2answers
216 views

Conditions for a group to be lattice-ordered

Given a set $S$ with a group operation $\cdot$ and a lattice ordering $\leq$, I wish to know when we can say that $\cdot$ preserves $\leq$, i.e. $(x\vee y)z=xz\vee yz$ and similarly for meets. ...
2
votes
2answers
188 views

When is a filter generated by a (countable) chain?

In any partial order $(P,\leq)$ it is easy to see that every chain generates (i.e., by taking the upwards closure) a filter, and any filter built as a result of the Rasiowa-Sikorski lemma in forcing ...
6
votes
1answer
313 views

A characterization of the poset of filters on a set

For the lattices of all subsets of a given set it is known an axiomatic characterization: A poset is isomorphic to a set of all subsets of some set iff it is a complete atomic boolean algebra. The ...
10
votes
3answers
162 views

Maximal chains in a quasi-order of linear order types

Let $\mathcal{T}_\kappa$ be the set of all linear order types of cardinality $\kappa$. Let $\prec$ denote a binary relation on $\mathcal{T}_\kappa$ representing embeddability of order types (note that ...
7
votes
1answer
265 views

What kind of category is a cyclically ordered set?

Background: A preorder is a binary relation $\leq$ which is reflexive and transitive. We can write the transitive property as ${\leq}(a,b)\wedge{\leq}(b,c)\to{\leq}(a,c)$. There are additional axioms ...
0
votes
1answer
286 views

Lattice ordered group

Does there exist a lattice-ordered group of rational rank $1$? This is true for totally ordered group.
4
votes
4answers
413 views

Why do we choose the standard total order on the integers?

I understand why the set of natural numbers $\mathbb N = \{ 0, 1, 2, \cdots \}$ is equipped with a total order. Indeed, every monoid has a pre-order, where $$n' \succeq n \quad \mathrm{if~and~only~if} ...