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Oct
19
comment Lists of small lattices and posets
Peter Jipsen maintains some lists of small ordered algebraic structures, including a nice page/applet that generates Hasse diagrams of all finite lattices with at most 7 elements here. (Some of the links on that page seem to be broken right now. I emailed Peter to let him know. Hopefully they will be fixed soon.) There are also diagrams of all connected posets of size 4 and 5 here.
Oct
19
revised Lists of small lattices and posets
the title seemed only marginally related to the question
Oct
19
suggested suggested edit on Lists of small lattices and posets
Sep
30
awarded  Explainer
Sep
27
comment Lattice which is not bounded lattice
The set $(0,1) \subset \mathbb{R}$ with the usual $\leq$ order is another example. It is not bounded below (above) because there is no element in $(0,1)$ that is less (greater) than every other element.
Sep
22
comment A Course in Universal Algebra (Millennium edition), page 74
@TimLee No, the "Sub" tab draws the lattice of subalgebras of the currently selected algebra. If "Sub" showed a 2x2, that means you're still looking at the 2-element lattice. After following the steps I gave above, you will see two lines in the Algebras list at the bottom of the UACalc window. The first is lat2. The second is the free algebra over lat2 that you constructed using the Tasks menu. Now, from this list of Algebras, select "F(2) over lat2". You can then click the Edit tab and see what the multiplication table is for the free algebra, and draw it using Drawing-->Algebra.
Sep
20
answered What does a lattice of the direct power of the two-element chain look like?
Sep
19
comment A Course in Universal Algebra (Millennium edition), page 74
You may find it instructive to try out some small examples in the Universal Algebra Calculator (www.uacalc.org). After you launch the application, try, for example: File -> Built In Algs -> lat2. Then: Tasks -> Free Algebra. You will be prompted for the number if generators. Enter 2, then click "Drawing" and "Go". You will see a 2x2 lattice. When learning this stuff, UACalc can be useful for verifying pencil/paper calculations.
Sep
19
comment A Course in Universal Algebra (Millennium edition), page 74
You ask, "any example?" ...but you already gave an example inside the parentheses. In your example, K contains only the two element lattice L. The free algebra over K on two generators is a 2x2 lattice, which is not isomorphic to L, so it is not isomorphic to any lattice in K.
May
15
awarded  Yearling
Apr
9
comment H P S class operators and their inequalities
Perhaps people would be more willing to answer if you accepted, or at least commented on, the answers provided to your earlier questions, for example this one or this one or this one or this one. This isn't simply a matter of reputation points. No one wants to spend time posting answers if they have the feeling their answers will be ignored.
Mar
29
comment What do we call functions that are definable by expressions?
Eran is right. See also: page 68 and 69 of Burris and Sankappanavar.
Mar
23
revised Suggestions for a learning roadmap for universal algebra?
deleted 15 characters in body
Mar
23
answered Proving every totally ordered and bounded lattice set is a Heyting Algebra
Mar
23
comment Proving every totally ordered and bounded lattice set is a Heyting Algebra
Okay, I'll change my comment to an answer, so the question can be marked answered. By the way, I recently started learning Coq, so if you implement this proof, could you share it? Thanks!
Mar
7
comment Can the actual scope of “lattice theory” be summarized as “algebraic order theory”?
I see. That makes sense. Thanks for elaborating on the motivation.
Mar
7
comment Can the actual scope of “lattice theory” be summarized as “algebraic order theory”?
What do you mean by "scope", and what is your motivation? Semilattice theory is connected to lattice theory insofar as a semilattice is a reduct of a lattice. One could argue that the "scope" of lattice theory extends beyond "pure" lattice theory, since lattices show up so often when studying other mathematical structures (e.g., lattices of: subalgebras, congruence realtions, ideals, normal subgroups, partitions, etc). However, "pure" lattice theory might be confined to the abstract theory of lattices. I think scope is hard to define, and I'm not sure why you would want to define it.
Feb
18
revised Subdirect embedding of a quotient algebra
added 20 characters in body
Feb
14
comment complete lattice
What do you mean by continuous?
Feb
14
comment complete lattice
Why do you write "(finite) complete" instead of "finite (complete)"?