3.4. Algebraic terms and term algebras

Algebraic drafts

As the concept of term was only intuitively introduced in 1.5, let us now formalize the case of terms using a purely algebraic language (without logical symbols), called algebraic terms, as mathematical systems in a set theoretical framework.
For convenience, let us work with only one type (the generalization to many types is easy) and introduce a class of systems more general than terms, that we shall call drafts. Variables will have a special treatment, without adding them as constants in the language.

Given an algebraic language L, an L-draft will be an L'-system (D,D) where D⊂ (LDD, such that:

Let us denote ∀xOD, ΨD(x)=(s(x), lx) ∈ LD where s(x)∈L and lxDns(x). We can also denote sD(x)=s(x) to let sD be a function with domain OD.
Well-foundedness can be equivalently written in any of these forms

AOD, (∀xOD, Im lxAVDxA) ⇒ A=OD
AD, (VDAD*(LA) ⊂A) ⇒ A = D
AD, VDAD ⇒∃xOD\A, Im lxA
AOD, A≠∅ ⇒ ∃xA, A⋂ Im lx = ∅

The set of used variables of (D,D), those which effectively occur, is VD⋂⋃xOD Im lx. Unused variables can be added or removed in D while keeping D fixed (by changing DOD∪(⋃xOD Im lx)), so that their presence may be irrelevant.
A ground draft is a draft with no variable, i.e. VD=∅. Thus, ΨD: DLD and SubLD = {D}.
More generally a draft is ground-like if it has no used variable (Dom DLOD).

Sub-drafts and terms

Drafts do not have interesting stable subsets (by well-foundedness), but let us introduce another stability concept for them.
A subset AD is a sub-draft of D (or a co-stable subset of D) if, denoting OA = AOD and ΨA= ΨD|OA, we have (Im ΨALA), i.e. ∀xOA, Im lxA.
Indeed, it remains well-founded, as can be seen on the last formulation of well-foundedness.

Like with stable subsets, any intersection of sub-drafts is also a sub-draft; the sub-draft co-generated by a subset is the intersection of all sub-drafts that include it.
A term is a draft co-generated by a single element which is its root.
Moreover, any union of sub-drafts is also a sub-draft (which was not the case for sub-algebras because an operation with arity >1 whose arguments take values in different sub-algebras may give a result outside their union).

There is a natural order relation on each draft D defined by xy ⇔ x∈ (the term with root y). It is obviously a preorder. Its antisymmetry is less obvious; a proof for integers will be given in 3.6, while the general case will come from properties of well-founded relations in the study of Galois connections.

Categories of drafts

As particular relational systems, classes of L-drafts form concrete categories. Between two L-drafts D,E,

f ∈MorL(D,E) ⇔ (f[OD]⊂OE ∧ ΨEf|OD= fL০ΨD)

where the equality condition can be split as

sEf|OD = sD
xOD, lf(x)=flx

Another kind of category of drafts can be considered, with objects also L-drafts but with a common set of variables (VD=VE=V) and taking smaller sets of morphisms: the variables-preserving morphisms, i.e. moreover satisfying

f|V = IdV

But for any element t in any draft, the term T co-generated by {t} has as set of variables TV (which is the set of used variables of T unless T={t}⊂V) generally smaller than V, so the admission of terms defined as subsets co-generated by singletons in such a category requires this loosening of the condition.

This naturally simplifies when reformulating such categories as those of ground (LV)-drafts: in each draft, variable symbols are replaced (reinterpreted) by constant symbols added to the language, so ΨE is extended by IdV, to form a ground (LV)-draft.

Intepretations of drafts in algebras

For any L-draft D and any L-algebra E, an interpretation of D in E is a morphism f∈MorL(D,E), i.e. f|OD= φEfL০ΨD, which can also be written

xOD, f(x) = s(x)E(flx)

Theorem. For any L-draft D with set of variables V and any L-algebra E, any vEV is uniquely extensible to an interpretation of D:
∃!h∈MorL(D,E), h|V = v, equivalently ∃!hEOD, vh ∈MorL(D,E).

The uniqueness comes from a previous proposition.

Proof of existence.
S = {AD | VA∧ Im ΨALA}
vK = ⋃AS {f∈MorL(A,E) | f|V =v}
f,gK, B = Dom f ⋂ Dom g ⇒ (f|BKg|BK) ⇒ f|B=g|B
fK Gr f = Gr h
C= Dom h = ⋃fK Dom fS
hK
(CD*(LC) ∋ x↦ (xC ? h(x) : φE(hLD(x))))) ∈ K (see conditional operator)
D*(LC) ⊂ C
C=D

Term algebras

An L-algebra (EE) is called a term algebra if it is injective and 〈E\Im φEL = E. Thus it is also an L-draft with ΨE = φE-1. As such, it is ground if moreover E=Im φE.
So, a ground term algebra is an algebra both minimal and injective, and thus also bijective.
By the above interpretation theorem, in any variables-preserving category of L-drafts with a fixed set V of variables (category of ground (LV)-drafts), any term L-algebra F, when present, is a final object. Thus any 2 of them are isomorphic, by a unique variables-preserving isomorphism.
In particular, any ground term L-algebra is a final object in any category of ground L-drafts, and an initial object in any category of L-algebras.

If L does not contain any constant then ∅ is a ground term L-algebra.
If L only contains constants, then ground term L-algebras are the copies of L.

Proposition. For any ground term L-algebra K and any injective L-algebra M, the unique f∈MorL(K,M) is injective.

Proof 1. By a previous result, {xK | ∀yK, f(x) = f(y) ⇒ x=y} ∈ SubLK, thus = K.

Proof 2. The subalgebra Im f of M is both injective (subalgebra of an injective algebra) and minimal, thus a ground term L-algebra, and the morphism f between initial L-algebras K and Im f is an isomorphism.

Role of term algebras as sets of all terms

Any term algebra F plays the role of the "set of all terms" with its list V of variable symbols, for the following reason:
Each element of F bijectively defines a term in F as the sub-draft of F it co-generates, thus where it is the root.
For any L-term T with root t and variables ⊂V, the unique f∈Mor(T,F) such that f|TV = IdTV represents it in F as the term Imf with root f(t).
Then its interpretation in any L-algebra E extending any vEV, is determined by the unique g∈MorL(F,E) extending v, as gf∈Mor(T,E), with result g(f(t)).
The same for terms whose set of variables V' is interpreted in E by the composite of a function from V' to V, with one from V to E (instead of having V'V).
For any subset A of an L-algebra E, any term algebra FA whose set of variables is a copy of A, represents the set of all L-terms with variables interpreted in A. Then, the L-subalgebra 〈AL of E is the image of the interpretation of FA in E, i.e. the set of all values of these terms.

Preservation of operations defined by terms

Sets of morphisms between L-algebras E,F remain unchanged when adding to L any operation symbol defined by an L-term T with root t and variables among V :

MorL(E,F) ⊂ MorT(E,F) ⊂ Mor{t}(E,F)

Proof:
The interpretation of T as a set of V-ary algebraic symbols in each E is defined by ∀vEV, ∀tT, tE(v) = gv(t) where gv∈MorL(T,E) and g|TV=v.
Thus ∀f∈MorL(E,F), ∀vEV, (fgv∈MorL(T,F) ∧ (fgv)|TV = fv) ∴ tF(fv) = (fgv)(t) = f(tE(v)). ∎
This may be seen as a particular case of conservation of logically defined interpreted relations in categories of relational systems, since any term defining an operation can be re-expressed as a formula defining the graph of this operation, using logical symbols ∃ and ∧. The advantage now that it is established for the general case of abstractly conceived terms no matter their size, instead of concretely written terms on which the conservation property must be repeatedly used for each occurrence of symbol it contains.
Back to homepage : Set theory and foundations of mathematics
1. First foundations of mathematics
2. Set theory (continued)
3. Algebra 1
3.1. Morphisms of relational systems and concrete categories
3.2. Special morphisms
3.3. Algebras
3.4. Algebraic terms and term algebras
3.5. Integers and recursion
3.6. Arithmetic with addition
4. Model Theory
4.1. Finiteness and countability
4.2. The Completeness Theorem
4.3. Infinity and the axiom of choice
4.4. Non-standard models of Arithmetic
4.5. How theories develop
4.6. The Incompleteness Theorem