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1answer
150 views

Can nonisomorphic algebras generate isomorphic relatively free algebras?

Suppose that we have two varieties of algebras $A$ and $B$, whose operators all have arities less than some regular cardinal, and such that every $B$-algebra (please correct me if this is not the ...
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1answer
64 views

How to define the action of U(G) in this situation?

The usual action of $fg$ on $u⊗v$, where $f,g$ are elements in the Universal Enveloping Algebra $U(G)$ of a Lie algebra $G$ and $u,v$ are elements of a representation $V$ of $G$, is given by ...
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3answers
396 views

Can we represent all algebraic structures in First-Order logic?

Can we represent all algebraic structures in First-Order logic?
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1answer
183 views

Algebra terminology question

Suppose A is some algebraic structure, and x and y are two elements of the underlying set. Is there any more concise way of stating, "there exists an automorphism in A which maps x to y"?. It seems ...
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5answers
524 views

Constructing a counterexample in category theory

Exercise 10 in Geroch's Mathematical Physics asks whether direct products distribute over direct sums in arbitrary categories. (They do in the category of sets, which is what motivates the question). ...
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4answers
2k views

Simple explanation of a monad

I have been learning some functional programming recently and I so I have come across monads. I understand what they are in programming terms, but I would like to understand what they are ...