4
votes
2answers
142 views

Different ways of constructing the free group over a set.

This could be too broad if we're not careful. I'm sorry if it ends up that way. Let's put together a list of different constructions of the free group $F_X$ over a given set $X$. It seems to be ...
3
votes
1answer
108 views

Can I define a category as a monoid with partially defined multiplication?

A groupoid can either be thought of as a category whose morphisms are isomorphisms, or as a generalization of a group whose multiplication is only partially defined. Can I do a similar thing with ...
3
votes
1answer
104 views

Semilattices are congruence-semi-distributive

A semilattice $(S,\cdot)$ is a commutative idempotent semigroup. A congruence on a semilattice is an equivalence relation that preserves multiplication, i.e. $x_1\mathrel{\theta} y_1$ and ...
3
votes
1answer
82 views

Completing a Partially Defined Associative Binary Operation

This is more like a question about terminology. I would like to hear some recommendations of books that discuss algebraic structures with one partially defined associative binary operation, and the ...
-1
votes
2answers
105 views

Building factor semigroup w.r.t. $I_n = \{ w \in A^+ : |w| \ge n \}$.

I don't understand the following factor semigroup. Consider the pseudovariety $N$ of nilpotent semigroups. For any finite alphabet $A$, let $I_n = \{ w \in A^+ : |w| \ge n \}$. Then $A^+ / I_n$ is ...
4
votes
2answers
231 views

What is the smallest variety of algebras containing all fields?

A field is a ring whose nonzero elements form a commutative group under multiplication. A field is also a commutative inverse semigroup with respect to multiplication. The unique multiplicative ...
1
vote
2answers
207 views

Existence of universal enveloping inverse semigroup (similar to “Grothendieck group”)

Context In its simplest form, the Grothendieck group construction associates an abelian group to a commutative semigroup in a "universal way". Now I'm interested in the following nilpotent ...