All Questions
Tagged with order-theory abstract-algebra
245
questions
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37
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Galois connections give rise to complete lattices
I am reading Introduction to Lattices and Order, second edition, by Davey and Priestly. On page 161, it says
Every Galois connection $(^\rhd,^\lhd)$ gives rise to a pair of closure operators, $^{\rhd\...
- 1,889
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22
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If sub-universe $S$ of lattice has congruence $\theta$, does the lattice have a congruence $\lambda = \theta \cap S^2$? [duplicate]
Let $(L, \lor , \land )$ be a lattice and $S$ a sub-universe of the lattice. A sub-universe of a lattice will be any subset of the lattice set that is non-empty and closed under $\land$ and $\lor$. ...
- 3,468
1
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65
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simplicial category is generated by cofaces and codegeracies
As the title says, I'd like to understand whether the following proof of the well known fact that give $f \in \Delta([m],[n])$ weakly increasing is uniquely determined by being $f = \delta^{i_1}\circ \...
- 5,164
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1
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44
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Countability of a partially ordered set
First of all, thank you in advance to everyone who will look at my question, I saw a question yesterday and I thought of such a solution to the question, but I was not sure if it was my mistake or not,...
- 63
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1
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How to define a linear order on the set of $m$-ary operations on an $n$-element set?
Let $m$ and $n$ be positive integers, let $S$ be a linearly ordered $n$-element set, and consider the set of all $m$-ary operations on $S$. What is the most "natural" way to linearly order ...
- 21.5k
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23
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Listing the elements in a poset in "non-decreasing" order
In my problem, I have a specific poset which is also a lattice, but I think the result holds for any finite poset, so let $(P,\preceq)$ be a finite poset with $\#P=n$ and maximal element $M$ and ...
- 376
3
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52
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Subsets Whose Translates Under Group Action Are $\subseteq$-Ordered
Suppose $X$ is a space, and $G$ a topological group acting continuously upon it. I'm interested in those closed sets $A \subseteq X$ whose translates under each $g \in G$, up to equivalence, are ...
- 3,380
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51
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$\mathcal E$ is an algebra. Then, a set must be either partly E-dense or in $\mathcal E$?
$\mathcal E$ is an algebra on $[0,1]$. $E$ is in $\mathcal E$. A set $D$ is $\mathcal E$-dense if $D\cap E$ is nonempty for all positive measure $E$. We use notation $D$ for E-dense set. $D$ is not in ...
- 3,824
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75
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How many affine prime-quotient ultrafilters does a rational semiring have?
I know ultrafilters are considered powerful by more-learned mathematicians than I. I cannot profess to understand the reasons how and why although I can see the power of Zorn's Lemma and the axiom of ...
- 9,299
1
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1
answer
66
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The natural partial order on the set of idempotents in the semigroup of Boolean relation matrices.
Let $\mathcal{E}_n$ be the set of idempotents in the semigroup $\mathcal{B}_n$ of $n \times n$ Boolean relation matrices. The relation $E \leq F$ iff $EF=FE=E$ is called the natural partial order on $...
- 48
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1
answer
1k
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Minimal and maximal elements of a partially ordered set
I've been trying to comprehend the concept of minimal and maximal elements of a partially ordered set, but the explanations I find often involve the use of Hasse diagrams. Can someone please explain ...
- 152
5
votes
1
answer
94
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How to describe all semigroups $(S, \, \cdot)$ based on a choice operation?
Let $S$ be a non-empty set. We say that a binary operation $f \, \colon S \times S \to S$ is a choice operation if it always returns one of its arguments. In other words, $\forall \, a \in S \, \colon ...
- 1,815
2
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1
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48
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If $⊥$ is an operation on $X$ inversely compatibile at $x$ in $X$ with the order $\mathcal O$ on $X$ then is it directly compatible at $x$ too?
If $\bot$ is an operation on a set $X$ then we say that it is directly compatibile on the left with an order $\mathcal O$ on $X$ at any $x$ in $X$ if the inequality
$$
x_1\prec_\mathcal Ox_2
$$
with $...
- 4,906
5
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1
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100
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The successor of 0 in an ordered ring
The Setup
I am investigating properties of the following definition of ordered rings:
Let $R$ be a ring. We say $R$ is an ordered ring under the total order relation $\leq$ if, for all $a,b,c \in R$, ...
- 1,249
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100
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Multiplicative lattice with $(a \land b)\ast(a\lor b)=a \ast b$
The natural numbers $\mathbb{N}$ carry two (order-theoretic) lattice structures: One, say $L_1$, is the division lattice (where the join is the least common multiple and the meet is the greatest ...
- 1,769