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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 without variables) $c_1$, ..., $c_n$ the identity $t(c_1,...,c_n)=t'(c_1,...,c_n)$ follows from (the identities of) $T$ then so does $t(x_1,...,x_n)=t'(x_1,...,x_n)$.

In algebraic terms this means to characterize those varieties of algebras which are generated by their initial algebra (the free algebra on the empty set).

The only widely known example of this that I was able to come up with is the theory/variety of Boolean algebras. But in fact any algebra $A$ in any signature generates its own variety of this kind, by adding to the signature a bunch of constants in the well known way and then generating the subvariety by $A$ itself.

So I am interested in any "intrinsic" (say, category-theoretic) characterization of theories/varieties with the above property, as well as in any other familiar examples of such.

May one hope to actually classify such things up to, say, categorical equivalence?

Much later

After so much time I hardly made any progress on this, but here is one feature which might be useful.

I believe I have a (simple) proof that a variety is of the above kind if and only if every subdirectly irreducible algebra can be generated by no elements. That is, subdirectly irreducible quotients of the initial algebra are the only subdirectly irreducibles up to isomorphism.

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 without variables) $c_1$, ..., $c_n$ the identity $t(c_1,...,c_n)=t'(c_1,...,c_n)$ follows from (the identities of) $T$ then so does $t(x_1,...,x_n)=t'(x_1,...,x_n)$.

In algebraic terms this means to characterize those varieties of algebras which are generated by their initial algebra (the free algebra on the empty set).

The only widely known example of this that I was able to come up with is the theory/variety of Boolean algebras. But in fact any algebra $A$ in any signature generates its own variety of this kind, by adding to the signature a bunch of constants in the well known way and then generating the subvariety by $A$ itself.

So I am interested in any "intrinsic" (say, category-theoretic) characterization of theories/varieties with the above property, as well as in any other familiar examples of such.

May one hope to actually classify such things up to, say, categorical equivalence?

Much later

After so much time I hardly made any progress on this, but here is one feature which might be useful.

I believe I have a (simple) proof that a variety $\mathscr V$ is of the above kind if and only if all of its subdirectly irreducible algebras can be generated by no elements. That is, subdirectly irreducible quotients of the initial algebra of $\mathscr V$ are the only subdirectly irreducibles in $\mathscr V$ up to isomorphism.

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 without variables) $c_1$, ..., $c_n$ the identity $t(c_1,...,c_n)=t'(c_1,...,c_n)$ follows from (the identities of) $T$ then so does $t(x_1,...,x_n)=t'(x_1,...,x_n)$.

In algebraic terms this means to characterize those varieties of algebras which are generated by their initial algebra (the free algebra on the empty set).

The only widely known example of this that I was able to come up with is the theory/variety of Boolean algebras. But in fact any algebra $A$ in any signature generates its own variety of this kind, by adding to the signature a bunch of constants in the well known way and then generating the subvariety by $A$ itself.

So I am interested in any "intrinsic" (say, category-theoretic) characterization of theories/varieties with the above property, as well as in any other familiar examples of such.

May one hope to actually classify such things up to, say, categorical equivalence?

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 without variables) $c_1$, ..., $c_n$ the identity $t(c_1,...,c_n)=t'(c_1,...,c_n)$ follows from (the identities of) $T$ then so does $t(x_1,...,x_n)=t'(x_1,...,x_n)$.

In algebraic terms this means to characterize those varieties of algebras which are generated by their initial algebra (the free algebra on the empty set).

The only widely known example of this that I was able to come up with is the theory/variety of Boolean algebras. But in fact any algebra $A$ in any signature generates its own variety of this kind, by adding to the signature a bunch of constants in the well known way and then generating the subvariety by $A$ itself.

So I am interested in any "intrinsic" (say, category-theoretic) characterization of theories/varieties with the above property, as well as in any other familiar examples of such.

May one hope to actually classify such things up to, say, categorical equivalence?

Much later

After so much time I hardly made any progress on this, but here is one feature which might be useful.

I believe I have a (simple) proof that a variety is of the above kind if and only if every subdirectly irreducible algebra can be generated by no elements. That is, subdirectly irreducible quotients of the initial algebra are the only subdirectly irreducibles up to isomorphism.

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 without variables) $c_1$, ..., $c_n$ the identity $t(c_1,...,c_n)=t'(c_1,...,c_n)$ follows from (the identities of) $T$ then so does $t(x_1,...,x_n)=t'(x_1,...,x_n)$.

In algebraic terms this means to characterize those varieties of algebras which are generated by their initial algebra (i.~e. the free algebra on the empty set).

The only widely known example of this that I was able to come up with is the theory/variety of Boolean algebras. But in fact any algebra $A$ in any signature generates its own variety of this kind, by adding to the signature a bunch of constants in the well known way and then generating the subvariety by $A$ itself.

So I am interested in any "intrinsic" (say, category-theoretic) characterization of theories/varieties with the above property, as well as in any other familiar examples of such.

May one hope to actually classify such things up to, say, categorical equivalence?

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 without variables) $c_1$, ..., $c_n$ the identity $t(c_1,...,c_n)=t'(c_1,...,c_n)$ follows from (the identities of) $T$ then so does $t(x_1,...,x_n)=t'(x_1,...,x_n)$.

In algebraic terms this means to characterize those varieties of algebras which are generated by their initial algebra (the free algebra on the empty set).

The only widely known example of this that I was able to come up with is the theory/variety of Boolean algebras. But in fact any algebra $A$ in any signature generates its own variety of this kind, by adding to the signature a bunch of constants in the well known way and then generating the subvariety by $A$ itself.

So I am interested in any "intrinsic" (say, category-theoretic) characterization of theories/varieties with the above property, as well as in any other familiar examples of such.

May one hope to actually classify such things up to, say, categorical equivalence?