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Results tagged with universal-algebra
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user 16537
The study of algebraic structures and properties applying to large classes of such structures. For example, ideas from group theory and ring theory are extended and considered for structures with other signatures (systems of basic or fundamental operations).
8
votes
2
answers
541
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If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all:
Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups?
If yes, then $S$ is a subdirect product of subdirectly irreducible groups …
- 10.5k
7
votes
2
answers
459
views
Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative ...
By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ta …
- 10.5k
3
votes
Accepted
Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative ...
Sorry for answering my own question, but YCor's construction in a related thread (here) gave me a lightbulb moment. Hopefully, it's not a broken lightbulb.
The answer to the question asked in the OP …
- 10.5k
2
votes
1
answer
211
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Terminology for a monoid $(H, \cdot)$ s.t. $ax=a$ or $xa =a$ only if $x$ is a unit
Let $(H, \cdot)$ be a (multiplicative) monoid. Is there any consolidated name for the following Property $\text{(P)}$, or for the class of monoids for which it is satisfied?
$$\text{(P) If }\,xy = x …
- 10.5k
1
vote
Accepted
Terminology for a monoid $(H, \cdot)$ s.t. $ax=a$ or $xa =a$ only if $x$ is a unit
Sorry for answering my own question, but it's definitely clear that there is no consolidated terminology for the kind of properties mentioned in the OP. One reason could be that they have never been c …
- 10.5k