All Questions
Tagged with lattice-orders logic
54
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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
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Understanding the definition of congruences over a lattice
Let $(L, \land, \lor)$ a lattice and $\theta$ a binary relation over $L$. We say $\theta$ is a congruence iff
$$
x_0\theta x_1, y_0 \theta y_1 \Rightarrow (x_0 \lor y_0)\theta(x_1 \lor y_1)
$$
(and ...
- 3,468
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Number of lattices over a finite set
I'm interested in finding a general formula for the number of lattices possible over a finite set $S$ as a function of the set's cardinality.
For instance, how many lattices over $\{1, 2, 3\}$ are ...
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Example of a residuated prelinear lattice that isn't linear
A residuated lattice is an algebra
$$(L,\land, \lor, \star,\Rightarrow,0,1)$$
with four binary operations and two constants such that
$(L,\land,\lor,0,1)$ is a lattice with the largest element 1 and ...
- 49
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Reconstructing a closure operator from a set of fixed points
Let $L$ be a lattice, not necessarily complete. We define a closure operator as a function $f\colon L\to L$ which is:
idempotent, $f(f(x)) = f(x)$,
isotone, $x\leq y \Rightarrow f(x) \leq f(y)$,
...
- 386
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Is there a connection between the topology of $\mathbb{R}/\mathbb{Q}$ and the logic fundamental to Smooth Infinitesimal Analysis?
As I read through the discussion related to this question (Visualizing quotient groups: $\mathbb{R/Q}$) posted 11 years ago I am reminded of a line on page 20 in a book by J. L. Bell, A Primer of ...
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Boolean algebra is to classical logic like what is to relevant logic?
The Question:
Boolean algebra is to classical logic like what is to relevant logic?
Context:
I guess this is a terminology question, so there's not much I can add, except that I've been interested ...
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How does one go about "finding" the algebra of a particular logic?
My question is this: If I know a deductive system for a logic is there a simple way to derive the appropriate algebra for that system?
For example: I start with the natural deduction system for ...
- 418
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Determining if lattice elements are equal
I am working in a distributive lattice with top and bottom elements.
I would like to know if there is an algorithm to determine if $s=t$ for any two elements $s,t$ in the lattice. For example, if $t=s\...
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left-adjoint to join in a Heyting algebra
Define a Heyting algebra to be a bounded lattice $L$ with an operation $\to : L^{op}\times L \to L$ such that for any $x, a, b \in L$ we have $x\wedge a \leqslant b$ iff $x \leqslant a \to b$. ...
- 11k
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Algebraic closure for partial functions
Here's a definition (taken from here, p.9):
The $\leq$ relation is a partial order relation that stands for 'part of'. E.g. $x\leq x\oplus y$.
In another source, the same author illustrates this ...
user1048887
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Reading off semi-lattice diagram
I'm reading a chapter about mereology (in a handbook of linguistics), and I have some questions. A prepring is available here (p. 519). The symbol $\leq$ is to be interpreted as non-strict parthood, ...
user1048887
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Question on Logic
Consider the following (complete and distributive) three-element lattice $\mathbf{A}_3= \langle \{1,a,0\}, \wedge, \vee, \Rightarrow, ^*,1, 0 \rangle$ where $1^*=0, a^*=1, 0=1^*$ and $x \Rightarrow y =...
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L is a complete lattice, prove that there exists $x \in L$, which is a fixed point of $f$ and is the smallest $y$ that $f(y) \le y$.
I have such task:
Let $(L, \le)$ be a complete lattice and $f: L \to L$ a monotonic function. Proof that there exists $x \in L$ which is a fixed point of $f$ and is the smallest $y$ that $f(y) \le y$.
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Can a free complete lattice on three generators exist in $\mathsf{NFU}$?
Also asked at MO.
It's a fun exercise to show in $\mathsf{ZF}$ that "the free complete lattice on $3$ generators" doesn't actually exist. The punchline, unsurprisingly, is size: a putative ...
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