Varieties

Let L an algebraic language, and K an L-algebra generated by a finite subset BK. We shall say that an L-algebra M satisfies (K,B) if, equivalently,
uMB, ∃f∈MorL(K,M), f|B=u
f∈MorL(K,MMB),∀bB, f(b)=πb
In the category of L-algebras that satisfy (K,B), B is a basis of K (this f is necessarily unique because K is generated by B).
This condition is equivalent to : for any equality between terms found true in K where the list of free variables is B (or whose interpretation is a bijection with B), M also satisfies this equality written under universal quantifiers.
Indeed, the morphism f is defined by mapping any element of K, value of any term with variables in B, into the element of M defined by the same term; for this to be coherent, 2 terms with the same value in K should also have the same value in M.
Conversely, any formula made of universal quantifiers over an equality of terms t=t', is equivalent to the satisfaction of some (K,B), that is the quotient of the algebra of all terms with the same variables, by the relation of being obtainable from each other by replacements of occurrences of t and t' as subterms.
Except if it is a statement of equality between 2 variables; to avoid such exceptions we may need to modify our above definition by seeing B not as a subset of K but as a family of elements of K ; this essentially differs only when this family is not injective, but the M satisfying such conditions must be empty or singletons, which do not form a very interesting category).

Thus, these L-algebras are also K-algebras where K is seen as a set of symbols with a common set B of arguments (thus a common arity). The interpretation of K in an L-algebra M satisfying (K,B) is characterized (we may say "recursively defined") by the condition of the above f , namely (using similar notations to those of group action which are a particular case): ∀uMB,
Between any two L-algebras M,N that satisfy (K,B), any g∈MorL(M,N) is also a K-morphism because ∀kK, ∀uMB,
(f∈MorL(K,M) ∧ f|B=ukM(u)=f(k))
⇒ (gf∈MorL(K,N) ∧ (gf)|B=gug(kM(u)))=(gf)(k))
g(kM(u)) = kN(gu)

Conversely, any sL with the same arity (or even lower arity) has an image in K with the same interpretation in this category.
Namely, any xBns, gives it an image k = sK(x), that is "replacing the variables of s by those from B by the substitution x". Thus, for any L-algebra M satisfying (K,B),∀uMB, kM(u) = sM(ux).
If x is bijective then this substitution is inversible, so that the interpretations of k and s are just copies of each other.
If x is just injective, we can still redefine s from k by mapping the missing variables, i.e. in (B \ Im x), either to those of s or to constants in L : this can be done except if s was a constant and we are removing it as well as all constant symbols from L, therefore admitting Ø in the category, where no constant symbol can be interpreted.
So, in this category, the set K of operations "already contains" any symbol from L with the same arity.

The class of all L-algebras satisfying any given family of conditions (Ki,Bi)iI is called a variety.

If the family (Ki,Bi)iI encompasses all arities (at least those present in L), then we may as well forget L and reinterpret the L-algebras in this variety, as L'-algebras where L'=⋃iI Ki. However, two questions remain:

These questions will be positively answered below.

Some stability properties of varieties

We can easily verify that:
Any subalgebra of an L-algebra satisfying (K,B) also satisfies (K,B) (because K is generated by B).
Any product of L-algebras satisfying (K,B) also satisfies (K,B).

Now comes the hard stuff:

Theorem. For any L-algebra A and any variety V of L-algebras, the category of all (M,f) where M is an L-algebra in V and f ∈MorL(A,M), has an initial object (A',φ), called the quotient of A by the family of conditions defining the variety.

Proof.

Let H be the set of all equivalence relations h on A such that A/h is an L-algebra in V, with the canonical projection ph∈MorL(A,A/h).
Let φ =∏hH ph and A' = Im φ ⊂ P =∏hH A/h.
A' satisfies all (Ki,Bi)iI because it is a subalgebra of a product P of L-algebras that do.
To verify that (A',φ) is an initial object in this category, let (M,f) be another object. We need to show that there is a unique morphism g from (A',φ) to (M,f), i.e. g∈MorL(A',M) such that gφ=f.
In the work on quotients (2.9) we saw the existence of a unique g such that gφ=f, written g=f/φ with domain Im φ = A', provided that ~φ ⊂~f . The condition of uniqueness here is Im φ = A'. The existence is obtained as g=j০πh, where h = ~f ,  πh ∈MorL(A',A/h) is the restriction of the canonical projection of P on its factor A/h, and j = (f/h) ∈ MorL(A/h,M). Indeed,
g
φ = j০πhφ = jph = f.    ∎
Now we can answer the previous questions.
First, to replace (Ki,Bi) by some (K'i,B'i) where K'i satisfies all (Kj,Bj) : we just need to take as K'i the quotient of Ki by this family of conditions.
If the restriction of this quotient φ to Bi was not injective, then only trivial algebras (empty or singletons) would satisfy all conditions. Indeed:
If there is an L-algebra M with elements xx' and satisfying all conditions then for all bb' in B, ∃uMBi , u(b)=xu(b')=x' and ∃f∈MorL(Ki,M), f|B=u thus ∃g∈MorL(K'i,M), gφ=f, and φ(b)φ(b') because f(b)≠f(b').
Now assuming it injective, the arity is preserved; the verification of equivalence of the conditions (Ki,Bi) and (K'i,B'i) as predicates on the class of algebras satisfying all (Kj,Bj) for ji , is straightforward and left to the reader.

Now for fusing several conditions (K,B) and (K',B') with the same arity into one: just take the quotient (K",B") of (K,B) by the condition (K',B'). For the same reason as the previous question, any algebra satisfying both will satisfy (K",B").
Any M satisfying (K",B") will satisfy (K,B); let us verify (K',B'), still assuming that we have a bijection v from B" to B' :∀uMB', since M satisfies (K",B"), the map uvMB" is extensible as a morphism f ∈ MorL(K",M). Since (K",B") satisfies (K',B'), the map v-1 from B' to B" is extensible as g∈MorL(K',K"). Thus fg∈MorL(K',M) is an extension of u.∎
In fact, when B and B' are in bijection, the quotient of (K,B) by both conditions (K,B) and (K',B') (plus any list of other conditions), is isomorphic to the quotient of (K',B') by the same conditions.



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