# Tagged Questions

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).

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### Is HSP(A) = ISP(A) decidable?

Let $A$ be a finite algebra for some finitary signature. Is it decidable whether $\mathbb{H}\mathbb{S}\mathbb{P}(A) = \mathbb{I}\mathbb{S}\mathbb{P}(A)$? That is, whether the variety ...
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### Variety of commutative semi group [closed]

V is a variety of commutative semi group satisfying the identity $x^2 = x^3$. I need to prove that: $|F_V(\{x_1\dots,x_n\})|$ = $3^n -1$. Any hints on this ? $F_V$ is V-free algebra.
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### SHPS and SPHS inequality using monounary algebra

Let $A_n = \{(1,\ldots,n) , f \}$ where $f(i) = (i+1)$ if $i \neq n$ otherwise $f(n) = 1$. This describes a mono unary algebra. The proof for $HPS \neq SPHS$ I know uses metabelian groups and was ...
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### H S class operator and its equality

$A \in S(K)$ iff $A$ is a subalgebra of some member of $K$ $A \in H(K)$ iff $A$ is a homomorphic image of some member of $K$ It is trivial to see the containment $SH \leq HS$. Taking a simple ...
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### When does a cogenerator determine a variety?

Two varieties of universal algebras are categorically equivalent iff their respective full subcategories of finitely generated free algebras are equivalent. Roughly speaking, this follows because they ...
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### Minimal generating sets of free algebras of varieties

Let $V$ be a variety and $F$ be a relatively free algebra in $V$. Suppose $X$ is a minimal generating set for $F$. Under what conditions we can deduce that $X$ is a free basis of $F$?
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### Axiomatizing orientation in the complex plane

Lately I've begun to suspect that a certain ternary relation might play a role in $\bf{C}$ analogous to the role played by the binary relation $>$ in $\bf{R}$, namely, the relation that the ...
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### Locally finite varieties which are not finitely generated

Let $\Sigma$ be a signature consisting of operations with finite arity. Let $\mathcal{V}$ be a variety of algebras for this signature. Further suppose that $\mathcal{V}$ is locally finite i.e. every ...
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### Equational theories determined by “identities without variables”

How to characterize equational theories $T$ which have the following property: for any two terms $t(x_1,...,x_n)$ and $t'(x_1,...,x_n)$ in the signature of $T$, if for any closed terms (i. e. terms ...
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### What do algebraic theories with strictly terminal trivial models look like?

By algebraic theory I mean one in the sense of Lawvere, i.e. a collection of finitary operations, including projections, together with a multi-composition satisfying the obvious axioms. (I believe ...
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### Existence of a construction in Universal Algebra: infinite trees

Is anything known about the following construction? Fix a signature (function symbols with arities incl 0) Sigma and a Sigma-algebra A. Construct a new Sigma-algebra T(A) as follows: The carrier set ...
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### Finite generation of vector identities

This question is partially motivated by Looking for a comprehensive referece for vector identities, although that question may not be appropriate for MO. Consider the set $\mathcal{E}$ of all valid ...
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### Complete Boolean algebra not isomorphic to a $\sigma$-algebra

Does there exist a complete Boolean algebra that is not isomorphic to any $\sigma$-algebra? If so, what is an easy or canonical example or construction?
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### The Universal Algebra of a sigma-Algebra

I am searching for the 'dual' algebraic structure of a Sigma Algebra. The notion of duallity is like on the case of the Boolean Algebra and Set Algera. If X is a set, the complement and intersection ...
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### Product of two algebras with maximum condition

Suppose $A$ and $B$ are two algebras of the same signature, both having maximum condition on sub-algebras. Is it true that $A\times B$ has the same property?
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### Varieties where every algebra is free

I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...
By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes. Equivalently (i) every interpretation of ...