congruencer

This program checks holding lattice identity in lattice of congruences of partitions of a finite set.

Usage

Let`s start with example. In directory src you can see files with example tests. Consider one: src/test2

   1 2 3 4 5 6 7
s1 1 2 3 4 5 6 7
s2 1 1 1 4 4 6 6
s3 2 2 2 5 5 7 7
s4 3 3 3 5 5 6 6
s5 2 2 2 4 4 6 6

This file describes transition function of automaton or action function of an act. Meaning of rows:

• First line contain all elements of automaton (or act). Here it`s 7. Element name can consist of more than one letter, but without spaces.
• Next lines contains action on specific semigroup element. For example, second line means: 1*s1 = 1, 1*s2 = 2, etc.

If you don`t want specify some value of transition function, just print _ in the line like this: s3 2 _ 2 5 _ _ 7.

Compilation

If you want to recompile this, you need Haskell Stack.

On Windows or Linux for compile type:

stack clean && stack build

For run:

stack exec congruencer-exe -- test\test2 --help

Examples

1. If you execute following command: congruencer test2 -m - program generates lattice of congruences and check holding modular identity. Output:

Congruences: 30
Modularity: True

2. congruencer test2 -m -l gives:

Congruences: 30
graph lattice {
        rankdir = TB;
        ratio = 0.75;
        node[shape = none];
        "(1234567)" -- "(67)(12345)";
        "(1234567)" -- "(12367)(45)";
        "(1234567)" -- "(4567)(123)";
        "(4567)(123)" -- "(67)(45)(123)";
        "(4567)(123)" -- "(4567)(12)";
...
        "(45)(13)" -- "(45)";
        "(45)(12)" -- "(12)";
        "(45)(12)" -- "(45)";
        "(45)" -- "Delta";
        "(123)" -- "(12)";
        "(123)" -- "(13)";
        "(123)" -- "(23)";
        "(23)" -- "Delta";
        "(13)" -- "Delta";
        "(12)" -- "Delta";
}

(Some output skipped).

You can specify -o file.dot option and redirect this output to a file file.dot.

3. If you want just only print congruences, then execute: congruencer test2 -t. Here is out:

[(1234567),(4567)(123),(4567)(23),(4567)(13),(4567)(12),(4567),(12367)(45),(67)(12345),
(67)(45)(123),(67)(45)(23),(67)(45)(13),(67)(45)(12),(67)(45),(12367),(67)(123),(67)(23),
(67)(13),(67)(12),(67),(12345),(45)(123),(45)(23),(45)(13),(45)(12),(45),(123),(23),(13),(12),Delta]
Congruences: 30

For more info, specify flag -h.

My e-mail: [email protected]