1. Introduction

Model theory: 

In [1] we coined the concept of graph algebra, introduced graph terms and showed that graph term algebras are free graph algebras. There was, however, no model theory in the sense of traditional Universal Algebra. As a kind of “proof of concept”, we generalize therefore some basic model-theoretic concepts and results from traditional algebras to graph algebras. We concentrate on the concept “generated subalgebra” and the related problem of characterizing monomorphic and epimorphic homomorphisms, respectively.

To tackle this task, we make some substantial effort in Section 3 to revise and reformulate traditional set-theoretic definitions, constructions and proofs in Universal Algebra by means of more category-theoretic concepts and constructions. Relying on this reformulation, we can in Section 4 smoothly transfer concepts, definitions and results from traditional algebras to graph algebras. Especially, we prove that all monomorphic homomorphisms between graph algebras are regular.

In [1] we adapted the original idea of sketch operations [5] and defined the arity of a graph operation as a single graph inclusion. This definition does not allow us, however, to consider projections as legal graph operations. To be closer to the traditional concept of operation in Universal Algebra, and to be able to define an appropriate concept of derived graph operation, we declare therefore in this paper the arity of a graph operation as a span of graph inclusions. In Section 4.2 we clarify the relation between both versions and discuss to what extent they are equivalent.

To prove that all monomorphic homomorphisms between graph algebras are regular, we also introduce in Section 4.4 partial graph algebras. We define a so-called term completion procedure transforming partial graph algebras into total graph algebras. This procedure provides for each signature a free functor from the corresponding category of partial graph algebras to the corresponding category of graph algebras. The construction of graph term algebras turns out to be just a special case of this new procedure.

Derived graph operations: 

To understand why the strong interconnection between “representation of data” and “representation of derived operations” breaks down in the case of graph operations and to find out how to define derived graph operations in an appropriate way, we include in the paper a more in-depth analysis of the concept term and discuss in Section 3.4 substitution calculi in general.

In Section 3.7 we recall the construction of syntactic Lawvere theories as it is described in [8], for example. In Section 5.1 we discuss finite product categories and elucidate that terms can be characterized as normal forms for finite product expressions, thus Lawvere’s original slogan “composition is substitution” can be turned into the slogan “substitution is symbolic composition plus normalization”.

Reviewing the relationship between finite products and tensor products, we identify in Section 5.2 copying as the cause of the problem. We argue that, in case of graph operations, “copying of data”, as a computation of its own, has to be replaced by “soldering” of input and (!) output ports of computations.

In such a way, we end up in Section 5.3 with three mechanisms to construct new graph operations out of given ones: parallel composition, instantiation (“soldering” of input and output ports), and sequential composition.

Finally, we introduce in Section 5.4 graph operation expressions with a structure as close as possible to the structure of terms. We define their semantics, i.e., the derived graph operations we have been looking for, by means of the three mechanisms parallel composition, instantiation and sequential composition.

2. Notations and Preliminaries

Obviously, the category $\mathsf{Graph}$ is, by definition, isomorphic to the interpretation category $[\mathtt{MG}\to \mathsf{Set}]$ with $\mathtt{MG}$ the graph . On the other side, the adjunction $\mathbf{Pth}⊣\mathbf{Gr}$ ensures that for any small graph $\mathtt{G}$ the interpretation category $[\mathtt{G}\to \mathsf{C}]$ is isomorphic to the functor category $\left[\mathsf{P}\right(\mathtt{G})\to \mathsf{C}]$ and thus a presheaf topos.

• $dom(f;g):=\{x\in A\mid x\in dom(f),f(x)\in dom(g\left)\right\}$ and
• $f;g\left(a\right):=g\left(f\right(a\left)\right)$ for all $a\in dom(f;g)$.

3. Algebras and Term Algebras

3.1. Signatures, Algebras and Homomorphisms

To declare the arities of operation symbols we use canonical finite indexing sets

$$I_0=\varnothing ,I_n:=\{i_1,i_2,\dots ,i_n\}\mathrm{for}\mathrm{all}n\ge 1\mathrm{and}O=\left\{o\right\}.$$

(1)

For all $n\ge 2$ we assume $I_n$ to be equipped with a fixed total order $i_1<i_2<\dots <i_n$ thus we can reuse the tuple notation to represent maps $\mathbf{a}:I_n\to A$ as discussed in Section 2.

Definition 1

(Signature). A signature $\mathsf{\Sigma }=(OP,ar)$ is given by

• a set $OP$ ofoperation symbols,
• a map $ar$ assigning to each operation symbol $\mathtt{op}$ as its arity a pair $ar\left(\mathtt{op}\right)=(I_{\mathtt{op}},O_{\mathtt{op}})$ of finite sets with $I_{\mathtt{op}}=I_n$ for some $n\in \mathbb{N}$, and $O_{\mathtt{op}}=O=\left\{o\right\}$.

We say that $\mathtt{op}\in OP$ is ann-ary operation symbol if $I_{\mathtt{op}}=I_n$. If $I_{\mathtt{c}}=I_0$ for $\mathtt{c}\in OP$, we say also that $\mathtt{c}$ is aconstant symbol.

Remark 1

(Sets as Arities). There can be arbitrary many finite separeted inputs for an algebraic operation. We decide to work with explicit sets of names for the “input positions”. In contrast to possibly multiple inputs, it is usually assumed that an algebraic operation has exactly one single output. For conformity reasons we introduce also a name for the single output position. This brings us closer to graph algebras where the single-output paradigm will be given up. As well the input as the output arity of a graph operation can be an arbitrary finite graph (see Definition 15).

Definition 2

(Algebra). Let Σ be a signature. A Σ-algebra $\mathcal{A}=(A,O{P}^{\mathcal{A}})$ given by

• a set A, called the carrier of $\mathcal{A}$, and
• a family $O{P}^{\mathcal{A}}=({\mathtt{op}}^{\mathcal{A}}:A^{I_{\mathtt{op}}}\to A^{O_{\mathtt{op}}}\mid \mathtt{op}\in OP)$ of maps called $operations$.

We say that ${\mathtt{op}}^{\mathcal{A}}$ is ann-ary operation if $I_{\mathtt{op}}=I_n$. If $I_{\mathtt{c}}=I_0$, we say also that ${\mathtt{c}}^{\mathcal{A}}$ is a constant operation, or simply aconstant.

In the case where the signature $\mathsf{\Sigma }$ has no constant symbols, the empty set constitutes a $\mathsf{\Sigma }$-algebra, called the empty Σ-algebra.

Now we reformulate the traditional concept of homomorphism.

Definition 3

(Homomorphism). Let Σ be a signature. A $\mathsf{\Sigma }$-homomorphism $h:\mathcal{A}\to \mathcal{B}$ between two Σ-algebras $\mathcal{A}$ and $\mathcal{B}$ is a map $h:A\to B$ satisfying the following homomorphism condition

$$\left(\mathbf{HC}\right){\mathtt{op}}^{\mathcal{A}}\left(\mathit{a}\right);h={\mathtt{op}}^{\mathcal{B}}(\mathit{a};h)\mathrm{for}\mathrm{all}\mathtt{op}\in OP\mathrm{and}\mathrm{all}\mathit{a}\in A^{I_{\mathtt{op}}}$$

For any sets A, B, I each map $h:A\to B$ induces a map $\_;h:A^I\to B^I$ thus we can, more abstractly but equivalently, express the homomorphism condition (HC) by the requirement that the above right square of maps commutes. Note, that in case of constant symbols $\mathtt{c}\in OP$, $I_{\mathtt{c}}=I_0$ the homomorphism condition turns into the equation ${\mathtt{op}}^{\mathcal{A}}\left(\right);h={\mathtt{op}}^{\mathcal{B}}\left(\right)$ if we apply our conventions in Section 2 concerning the tuple notation.

Given any $\mathsf{\Sigma }$-algebra $\mathcal{A}$, the identity map on the carrier set $i{d}_A:A\to A$ induces an identity $\mathsf{\Sigma }$-homomorphism $i{d}_{\mathcal{A}}:\mathcal{A}\to \mathcal{A}$. Similarly, given any $\mathsf{\Sigma }$-homomorphisms $f:\mathcal{A}\to \mathcal{B}$, $g:\mathcal{B}\to \mathcal{C}$, the composition $f;g:A\to C$ of the underlying maps induces a $\mathsf{\Sigma }$-homomorphism $f;g:\mathcal{A}\to \mathcal{C}$. This defines the category $\mathsf{Alg}\left(\mathsf{\Sigma }\right)$ with all $\mathsf{\Sigma }$-algebras as objects and all $\mathsf{\Sigma }$-homomorphisms as morphisms.

Proposition 1

(Forgetful Functor). The assignments $\mathcal{A}\mapsto A$ and $(f:\mathcal{A}\to \mathcal{B})\mapsto (f:A\to B)$ define a faithful forgetful functor

$${\mathbf{U}}_{\mathsf{\Sigma }}:\mathsf{Alg}\left(\mathsf{\Sigma }\right)\to \mathsf{Set}.$$

(2)

Characteristic for any incarnation of the concept algebra is that the corresponding categories of algebras inherit all limits from the respective underlying category. For the abstract concept of $\mathbf{F}$-algebra for an arbitrary functor $\mathbf{F}:\mathsf{C}\to \mathsf{C}$, for example, the category of all $\mathbf{F}$-algebras inherits all limits from the category $\mathsf{C}$ [11].

It is well-known that the category $\mathsf{Alg}\left(\mathsf{\Sigma }\right)$ inherits all limits from the category $\mathsf{Set}$. Following our methodological intention to lift things up to a more categorical element-free level, we demonstrate that the decision to work with sets of maps $A^I$ instead of Cartesian products $A^n$ enables us to give a pure categorical concise proof of this classical result. Since we want to do all later constructions and argumentations within $\mathsf{Set}$, we restrict ourselves to small limits.

Theorem 1

(Limits). $\mathsf{Alg}\left(\mathsf{\Sigma }\right)$ inherits any small limit from the category $\mathsf{Set}$, i.e., the functor ${\mathbf{U}}_{\mathsf{\Sigma }}:\mathsf{Alg}\left(\mathsf{\Sigma }\right)\to \mathsf{Set}$ reflects small limits. $\mathsf{Alg}\left(\mathsf{\Sigma }\right)$ has therefore all small limits since $\mathsf{Set}$ does.

Proof. 

Let $\mathtt{J}$ be a small graph and $\delta :\mathtt{J}\to \mathsf{Alg}\left(\mathsf{\Sigma }\right)$ be a diagram in $\mathsf{Alg}\left(\mathsf{\Sigma }\right)$ where ${\delta }_v={\mathcal{A}}_v=(A_v,O{P}^{{\mathcal{A}}_v})$ for all vertices v in ${\mathtt{J}}_V$. We have to show that any limit cone $\pi :L\Rightarrow \delta ;{\mathbf{U}}_{\mathsf{\Sigma }}$ over the translated diagram $\delta ;{\mathbf{U}}_{\mathsf{\Sigma }}:\mathtt{J}\to \mathsf{Set}$ induces a limit cone $\pi :\mathcal{L}\Rightarrow \delta$ over $\delta$ in $\mathsf{Alg}\left(\mathsf{\Sigma }\right)$ such that $\mathcal{L}$ is a $\mathsf{\Sigma }$-algebra with L as its carrier.

To define the operations in $\mathcal{L}$, we note that for any operation symbol $\mathtt{op}$ in $OP$ and any map $\mathbf{l}:I_{\mathtt{op}}\to L$ we get a commutative cone $\mathbf{l};\pi :I_{\mathtt{op}}\Rightarrow \delta ;{\mathbf{U}}_{\mathsf{\Sigma }}$ in $\mathsf{Set}$ with ${(\mathbf{l};\pi )}_v:=\mathbf{l};{\pi }_v:I_{\mathtt{op}}\to A_v$ for all v in ${\mathtt{J}}_V$. Applying the respective operations ${\mathtt{op}}^{{\mathcal{A}}_v}$ to the maps $\mathbf{l};{\pi }_v$ gives us a new cone ${\mathtt{op}}^{\delta }(\mathbf{l};\pi ):O_{\mathtt{op}}\Rightarrow \delta ;{\mathbf{U}}_{\mathsf{\Sigma }}$ in $\mathsf{Set}$ with ${\mathtt{op}}^{\delta }{(\mathbf{l};\pi )}_v:={\mathtt{op}}^{{\mathcal{A}}_v}(\mathbf{l};{\pi }_v):O_{\mathtt{op}}\to A_v$ for all vertices v in ${\mathtt{J}}_V$. Now, for any edge $e:v\to w$ in ${\mathtt{J}}_E$, we have that ${\delta }_e:{\delta }_v\to {\delta }_w$ is a $\mathsf{\Sigma }$-homomorphism from ${\mathcal{A}}_v$ to ${\mathcal{A}}_w$ and thus, by the homomorphism condition and commutativity of $\pi :L\Rightarrow \delta ;{\mathbf{U}}_{\mathsf{\Sigma }}$ we have

$${\mathtt{op}}^{{\mathcal{A}}_v}(\mathbf{l};{\pi }_v);{\delta }_e={\mathtt{op}}^{{\mathcal{A}}_w}(\mathbf{l};{\pi }_v;{\delta }_e)={\mathtt{op}}^{{\mathcal{A}}_w}(\mathbf{l};{\pi }_w)$$

(3)

which encapsulates that the cone ${\mathtt{op}}^{\delta }(\mathbf{l};\pi ):O_{\mathtt{op}}\Rightarrow \delta ;{\mathbf{U}}_{\mathsf{\Sigma }}$ is commutative. By the universal property of $\pi$, there is a unique map $k:O_{\mathtt{op}}\to L$ which satisfies

$$k;{\pi }_v={\mathtt{op}}^{{\mathcal{A}}_v}(\mathbf{l};{\pi }_v)$$

(4)

for all v in ${\mathtt{J}}_V$. By defining ${\mathtt{op}}^{\mathcal{L}}\left(\mathbf{l}\right):=k$, we ensure that each map ${\pi }_v:L\to A_v$ induces a $\mathsf{\Sigma }$-homomorphism ${\pi }_v:\mathcal{L}\to {\mathcal{A}}_v$ for all v in ${\mathtt{J}}_v$. Thus, we get indeed a commutative cone $\pi :\mathcal{L}\Rightarrow \delta$ in $\mathsf{Alg}\left(\mathsf{\Sigma }\right)$.

It remains to show that $\pi :\mathcal{L}\Rightarrow \delta$ is a limit cone, i.e. for any other commutative cone $p:\mathcal{X}\Rightarrow \delta$ in $\mathsf{Alg}\left(\mathsf{\Sigma }\right)$, we have to show that there is a $\mathsf{\Sigma }$-homomorphism $\kappa :\mathcal{X}\to \mathcal{L}$ such that $\kappa ;{\pi }_v=p_v$ for all v in ${\mathtt{J}}_V$. Note that p induces a commutative cone $p;{\mathbf{U}}_{\mathsf{\Sigma }}:X\Rightarrow \delta ;{\mathbf{U}}_{\mathsf{\Sigma }}$ in $\mathsf{Set}$ with ${(p;{\mathbf{U}}_{\mathsf{\Sigma }})}_v:=p_v:X\to A_v$ for all v in ${\mathtt{J}}_V$. As $\pi$ is a limit cone over $\delta ;{\mathbf{U}}_{\mathsf{\Sigma }}$, there exists a unique map $\kappa :X\to L$ such that $\kappa ;{\pi }_v=p_v$ for all v in ${\mathtt{J}}_V$. We claim that $\kappa$ extends to the desired $\mathsf{\Sigma }$-homomorphism, by showing that

$${\mathtt{op}}^{\mathcal{X}}\left(\mathbf{x}\right);\kappa ={\mathtt{op}}^{\mathcal{L}}(\mathbf{x};\kappa )$$

(5)

for any $\mathtt{op}$ in $OP$ and $\mathbf{x}:I_{\mathtt{op}}\to X$. By definition, ${\mathtt{op}}^{\mathcal{L}}(\mathbf{x};\kappa )$ is the unique map such that ${\mathtt{op}}^{\mathcal{L}}(\mathbf{x};\kappa );{\pi }_v={\mathtt{op}}^{{\mathcal{A}}_v}(\mathbf{x};\kappa ;{\pi }_v)$ holds for all v in ${\mathtt{J}}_V$. Indeed, the map ${\mathtt{op}}^{\mathcal{X}}\left(\mathbf{x}\right);\kappa :O_{\mathtt{op}}\to X$ also satisfies this equality for all v in ${\mathtt{J}}_V$ as

$${\mathtt{op}}^{\mathcal{X}}\left(\mathbf{x}\right);\kappa ;{\pi }_v={\mathtt{op}}^{\mathcal{X}}\left(\mathbf{x}\right);p_v={\mathtt{op}}^{{\mathcal{A}}_v}(\mathbf{x};p_v)={\mathtt{op}}^{{\mathcal{A}}_v}(\mathbf{x};\kappa ;{\pi }_v).$$

By uniqueness of mediating morphisms we get (5), which shows that $\kappa :X\to L$ is indeed a $\mathsf{\Sigma }$-homomorphism from $\kappa :\mathcal{X}\to \mathcal{L}$. Moreover, it is the unique homomorphism such that $\kappa ;{\pi }_v=p_v$ and thus, $\pi :\mathcal{L}\Rightarrow \delta$ is a limit cone. □

Remark 2

(Hom-sets). The proof of Theorem 1 is based on the convention in Section 2 that we consider $A^I$ as a shorthand notation for the collection (hom-set) $\mathsf{Set}(I,A)$ of all morphisms in $\mathsf{Set}$ from I to A. $\mathsf{Set}(I,A)$ is a set since $\mathsf{Set}$ is locally small. $\mathsf{Set}(I,A)$ is isomorphic to a corresponding exponential object in $\mathsf{Set}$, but this isomorphism does not play any role in the paper.

3.2. Subalgebras

As in the traditional approach, we can define subalgebras by means of set inclusions.

Definition 4

(Subalgebras). Let $\mathsf{\Sigma }=(OP,ar)$ be a signature. A Σ-algebra $\mathcal{A}=(A,O{P}^{\mathcal{A}})$ is a $\mathsf{\Sigma }$-subalgebra of a Σ-algebra $\mathcal{B}=(B,O{P}^{\mathcal{B}})$, $\mathcal{A}⊑\mathcal{B}$ in symbols, if $A\subseteq B$ and for all $\mathtt{op}\in OP$ and $\mathbf{a}:I_{\mathtt{op}}\to A$ the following diagram commutes:

Here ${\iota }_{A,B}:A\hookrightarrow B$ is the corresponding inclusion map from A into B.

A comparison of Definition 3 and Definition 4 makes, however, obvious that we can also simply describe $\mathsf{\Sigma }$-subalgebras as special kinds of $\mathsf{\Sigma }$-homomorphisms.

Corollary 1

(Subalgebras as Inclusion Homomorphisms). For Σ-algebras $\mathcal{A}$ and $\mathcal{B}$ such that $A\subseteq B$ we have that $\mathcal{A}$ is a Σ-subalgebra of $\mathcal{B}$ if, and only if, the inclusion map ${\iota }_{A,B}:A\hookrightarrow B$ establishes a Σ-homomorphism ${\iota }_{A,B}:\mathcal{A}\hookrightarrow \mathcal{B}$.

Since one of our objectives is to lift the traditional set-theoretic exposition of Universal Algebra to a more category-theoretic one, we will use, from now on, the concepts “$\mathsf{\Sigma }$-subalgebra” and “inclusion $\mathsf{\Sigma }$-homomorphism” interchangeably.

We know that the monomorphisms (epimorphisms) in $\mathsf{Set}$ are exactly the injective (surjective) maps, respectively. Any faithful functor reflects monomorphisms and epimorphisms. The forgetful functor ${\mathbf{U}}_{\mathsf{\Sigma }}:\mathsf{Alg}\left(\mathsf{\Sigma }\right)\to \mathsf{Set}$ is faithful thus we obtain

Corollary 2

(Injective and surjective Homomorphisms). If the underlying map $f:A\to B$ of a Σ-homomorphism $f:\mathcal{A}\to \mathcal{B}$ is injective (surjective) then $f:\mathcal{A}\to \mathcal{B}$ is a monomorphism (epimorphism) in $\mathsf{Alg}\left(\mathsf{\Sigma }\right)$.

In the traditional set-theoretic approach to Universal Algebra, a preferred tool to describe, construct and reason about subalgebras are subsets of the carrier which are closed w.r.t. to applications of the operations in the algebra.

Definition 5

(Closedness). Let $\mathcal{B}=(B,O{P}^{\mathcal{B}})$ be a Σ-algebra. We say a subset $A\subseteq B$ isclosed in $\mathcal{B}$if for all $\mathtt{op}\in OP$ and $\mathbf{a}:I_{\mathtt{op}}\to A$ the result ${\mathtt{op}}^{\mathcal{B}}(\mathbf{a};{\iota }_{A,B}):O_{\mathtt{op}}\to B$ of applying the operation ${\mathtt{op}}^{\mathcal{B}}$ in $\mathcal{B}$ to the input $\mathbf{a};{\iota }_{A,B}:I_{\mathtt{op}}\to B$ factors through the inclusion map ${\iota }_{A,B}:A\hookrightarrow B$, i.e., there exists a map $r_{\mathbf{a}}:O_{\mathtt{op}}\to B$ such that the following diagram commutes:

If $\mathcal{A}$ is a $\mathsf{\Sigma }$-subalgebra of $\mathcal{B}$ then the carrier A of $\mathcal{A}$ is obviously closed w.r.t. all the operations in $\mathcal{B}$. On the other side, the inclusion map ${\iota }_{A,B}:A\hookrightarrow B$ is a monomorphism in $\mathsf{Set}$. Therefore the map $r_{\mathbf{a}}:O_{\mathtt{op}}\to B$ in Definition 5 is unique if it exists. In such a way, the assignments $\mathbf{a}\mapsto r_{\mathbf{a}}$ define a total operation from $A^{I_{\mathtt{op}}}$ to $A^{O_{\mathtt{op}}}$ if A is closed, and we obtain the following result.

Proposition 2

(Subalgebra ≅ Closed Subset). There is a one-to-one correspondence between Σ-subalgebras of $\mathcal{B}$ and closed subsets of $\mathcal{B}$.

Proposition 2 suggests that there may be actually no need for the auxiliary concept closed subset in a more category-theoretic approach. This conjecture is supported by the observation that we can reconstruct the standard result that closed subsets are closed w.r.t. intersection, reformulated in terms of inclusion homomorphisms, as a special case of Theorem 1. To see this, we have to realize that the intersection of subsets can be described as a special limit construction, namely multiple pullbacks, in $\mathsf{Set}$.

Remark 3

(Multiple Pullbacks). Let I be a set and M be an I-indexed family $(A_i\subseteq B\mid i\in I)$ of subsets of a set B. We can describe this situation by a diagram $\delta :\mathtt{MP}\left(I\right)\to \mathsf{Set}$ with $\mathtt{MP}\left(I\right)$ a small graph given by $\mathtt{MP}{\left(\mathtt{I}\right)}_V:=I\cup \{*\}$, $\mathtt{MP}{\left(\mathtt{I}\right)}_E:=\{e_i:i\to *\mid i\in I\}$, and δ defined by ${\delta }_i:=A_i$ for all $i\in I$, ${\delta }_{*}:=B$ and inclusion maps ${\delta }_{e_i}:={\iota }_{A_i,B}:A_i\hookrightarrow B$ for all $i\in I$.

It is well-known and straightforward to prove that the intersection $\cap M:={\bigcap }_{i\in I}{A}_i$ together with the inclusion maps ${\iota }_{\cap M,A_i}:\cap M\hookrightarrow A_i$, $i\in I$ and ${\iota }_{\cap M,B}={\iota }_{\cap M,A_i};{\iota }_{A_i,B}:\cap M\hookrightarrow B$ is a limit cone of the diagram $\delta :\mathtt{MP}\left(I\right)\to \mathsf{Set}$ in $\mathsf{Set}$.

Limits of this shape are also calledmultiple pullbacks and they reflect monomorphisms: For any category $\mathsf{C}$, any diagram $\delta :\mathtt{MP}\left(I\right)\to \mathsf{C}$ and any limit cone $p:L\Rightarrow \delta$ all the morphisms $p_i:L\to {\delta }_i$, $i\in I$ are monomorphisms in $\mathsf{C}$ as as long as all the morphisms ${\delta }_{e_i}:{\delta }_i\to {\delta }_{*}$, $i\in I$ are.

Due to Remark 3 we can now replace and enhance the traditional statement

“If M is a family of closed subsets in $\mathcal{B}$, then its intersection $\cap M$ is closed as well”

by the following corollary of Theorem 1.

Corollary 3

(Intersection of Subalgebras). For any set I, any Σ-algebra $\mathcal{B}$, and any diagram $\delta :\mathtt{MP}\left(I\right)\to \mathsf{Alg}\left(\mathsf{\Sigma }\right)$ of Σ-subalgebras ${\delta }_{e_i}={\iota }_{A_i,B}:{\mathcal{A}}_i\hookrightarrow \mathcal{B}$, $i\in I$ of $\mathcal{B}$ there is a unique Σ-subalgebra $\mathcal{L}=(L,O{P}^{\mathcal{L}})$ of $\mathcal{B}$ with $L={\bigcap }_{i\in I}{A}_i$ that is a Σ-subalgebra of ${\mathcal{A}}_i$ for all $i\in I$.

Moreover, the inclusion Σ-homomorphisms ${\iota }_{L,A_i}:\mathcal{L}\hookrightarrow {\mathcal{A}}_i$, $i\in I$ and ${\iota }_{L,B}:\mathcal{L}\hookrightarrow \mathcal{B}$ constitute a multiple pullback, i.e., a limit cone of the diagram $\delta :\mathtt{MP}\left(I\right)\to \mathsf{Alg}\left(\mathsf{\Sigma }\right)$ in $\mathsf{Alg}\left(\mathsf{\Sigma }\right)$.

We call $\mathcal{L}=(L,O{P}^{\mathcal{L}})$ also the intersection of the I-indexed family $\mathcal{M}=({\mathcal{A}}_i\mid i\in I)$ of Σ-subalgebras of $\mathcal{B}$ and may use the notations $\bigcap \mathcal{M}$, ${\bigcap }_{i\in I}{\mathcal{A}}_i$ or, simply, $\bigcap {\mathcal{A}}_i$ to denote $\mathcal{L}$.

Traditionally, the $\mathsf{\Sigma }$-subalgebra $\mathcal{R}(A,\mathcal{B})$ of a $\mathsf{\Sigma }$-algebra $\mathcal{B}$generated by a subset $A\subseteq B$ can be defined as the $\mathsf{\Sigma }$-subalgebra with the carrier $R(A,\mathcal{B})$ constructed as the intersection of the following family of closed sets in $\mathcal{B}$:

$$R(A,\mathcal{B}):=\cap \{X\subseteq B\mid X\mathrm{closed}\mathrm{in}\mathcal{B}\mathrm{and}A\subseteq X\}.$$

(6)

Since, the collection of all subsets of a set B is a set as well, this definition matches the pattern of Corollary 3. We only have to choose for I the set $\{X\subseteq B\mid X\mathrm{closed}\mathrm{in}\mathcal{B}\mathrm{and}A\subseteq X\}$ itself or any isomorphic set.

Using this sleight of hand, we can take full advantage of the universal property of the intersection of subalgebras in $\mathsf{Alg}\left(\mathsf{\Sigma }\right)$, as stated in Corollary 3, and lift up the concept “generated by a subset” to the concept “accessible via a map”.

Definition 6

(Subalgebra accessible via a Map). For any Σ-algebra $\mathcal{B}$ and any map $f:A\to B$ let $\mathcal{M}$ be the set of all Σ-subalgebras $\mathcal{X}$ of $\mathcal{B}$ such that f factors through the inclusion map ${\iota }_{X,B}:X\hookrightarrow B$, i.e., there exists a map $f_{\mathcal{X}}:A\to X$ such that $f_{\mathcal{X}};{\iota }_{X,B}=f$.

We denote by $\mathcal{R}(f,\mathcal{B})$ the intersection of $\mathcal{M}$, according to Corollary 3. Especially, the carrier of $\mathcal{R}(f,\mathcal{B})$ is the intersection $R(f,\mathcal{B}):=\bigcap \{X\mid \mathcal{X}\in \mathcal{M}\}$ of sets. We call $\mathcal{R}(f,\mathcal{B})$ the$\mathsf{\Sigma }$-subalgebra of $\mathcal{B}$ accessible (reachable) via for thehomomorphic image of A w.r.t. f.

In case of inclusion maps $f={\iota }_{A,B}:A\hookrightarrow B$ we also use the traditional notation $\mathcal{R}(A,\mathcal{B})$ instead of $\mathcal{R}({\iota }_{A,B},\mathcal{B})$ and also call $\mathcal{R}(A,\mathcal{B})$ the$\mathsf{\Sigma }$-subalgebra of $\mathcal{B}$ generated by A.

Note, that the map $f_{\mathcal{X}}:A\to X$ in Definition 6 is unique, if it exists, since the inclusion map ${\iota }_{X,B}:X\hookrightarrow B$ is a monomorphism in $\mathsf{Set}$.

Corollary 4

(Homomorphic image includes Image). For any Σ-algebra $\mathcal{B}$ and any map $f:A\to B$ we have $f\left(A\right)\subseteq R(f,\mathcal{B})$ for the(set-theoretic) image $f\left(A\right):=\left\{f\right(a)\mid a\in A\}$ of A w.r.t. the map f.

Proof. 

Follows immediately from the observation that $f\left(A\right)=\bigcap \{Y\mid Y\in \mathcal{N}\}$ for the set $\mathcal{N}$ of all subsets Y of B such that f factors through the inclusion map ${\iota }_{Y,B}:Y\hookrightarrow B$ and that $\{X\mid \mathcal{X}\in \mathcal{M}\}\subseteq \mathcal{N}$ due to the definition of $\mathcal{M}$ and $\mathcal{R}(f,\mathcal{B})$ in Definition 6 and the definition of $\mathcal{N}$. □

Remark 4

(Well-powered). In category theory the adjective well-powered is used for categories $\mathsf{C}$ where for all objects A in $\mathsf{C}$ the collection of all subobjects of A is a set.

Of course, we do have the traditional concept of generated algebra and corresponding results available.

Definition 7

(Accessible and Generated Algebras). Let $\mathcal{B}$ be a Σ-algebra.

• $\mathcal{B}$ isaccessible via a map $f:A\to B$if$\mathcal{R}(f,\mathcal{B})=\mathcal{B}$.
• If $\mathcal{B}$ is accessible via an inclusion map ${\iota }_{A,B}:A\to B$, i.e., if$\mathcal{R}(A,\mathcal{B})=\mathcal{R}({\iota }_{A,B},\mathcal{B})=\mathcal{B}$, we say also that $\mathcal{B}$ isgenerated by A.
• $\mathcal{B}$ is said to be generated if it is generated by the empty set, i.e., accessible via the unique map ${\iota }_{\varnothing ,B}:\varnothing \hookrightarrow B$ from the initial object ∅ in $\mathsf{Set}$ to B.

Corollary 5.

A Σ-algebra $\mathcal{B}$ is generated if, and only if, there are no proper Σ-subalgebras of $\mathcal{B}$.

Corollary 6.

If a signature Σ has no constant symbols, then the empty Σ-algebra is the only generated Σ-algebra.

The concept accessible via a map can be utilized to find a characterization of epimorphisms in $\mathsf{Alg}\left(\mathsf{\Sigma }\right)$. First, we observe that “accessible via a map” implies “epic”.

Lemma 1

(Accessible implies Epic). A Σ-homomorphism $f:\mathcal{A}\to \mathcal{B}$ is an epimorphism in $\mathsf{Alg}\left(\mathsf{\Sigma }\right)$ if $\mathcal{B}$ is accessible via the underlying map $f:A\to B$, i.e., if $\mathcal{B}=\mathcal{R}(f,\mathcal{B})$.

Proof. 

We consider arbitrary Σ-homomorphisms $g,h:\mathcal{B}\to \mathcal{C}$ such that $f;g=f;h$.

We know that the set $X=\{b\in B\mid g(b)=h(b\left)\right\}\subseteq B$ together with the inclusion map ${\iota }_{X,B}:X\hookrightarrow B$ is an equalizer of the maps $g,h:B\to C$ in $\mathsf{Set}$. According to 1, there is a unique Σ-algebra $\mathcal{X}=(X,O{P}^{\mathcal{X}})$ such that ${\iota }_{X,B}:X\hookrightarrow B$ becomes an inclusion Σ-homomorphism ${\iota }_{X,B}:\mathcal{X}\hookrightarrow \mathcal{B}$ which is, moreover, the equalizer of the Σ-homomorphisms $g,h:\mathcal{B}\to \mathcal{C}$. Assumption $f;g=f;h$ ensures that there exists a unique map $f_{\mathcal{X}}:A\to X$ with $f_{\mathcal{X}};{\iota }_{X,B}=f$. Due to the construction of $\mathcal{R}(f,\mathcal{B})$ in Definition 6, we have an inclusion Σ-homomorphism ${\iota }_{R(f,\mathcal{B}),X}:\mathcal{R}(f,\mathcal{B})\hookrightarrow \mathcal{X}$. Accessibility of $\mathcal{B}$ means $\mathcal{B}=\mathcal{R}(f,\mathcal{B})$ thus we get, finally, $\mathcal{X}=\mathcal{B}$. This means, however, that the equalizer of g and h is the identity on B thus we have $g=h$ as required. □

To show that, on the other side, epic implies accessible we can take advantage of the following result.

Proposition 3

(Subalgebras are Regular Monos). For any Σ-subalgebra ${\iota }_{\mathcal{A},\mathcal{B}}:\mathcal{A}\hookrightarrow \mathcal{B}$ of a Σ-algebra $\mathcal{B}$ there exists a Σ-algebra $\mathcal{C}$ and parallel Σ-homomorphisms $g,h:\mathcal{B}\to \mathcal{C}$ such that ${\iota }_{\mathcal{A},\mathcal{B}}:\mathcal{A}\hookrightarrow \mathcal{B}$ is the equalizer of g and h.

Proof. 

We construct the pushout of the span of inclusion maps (see the left diagram below). We set $C:=B{+}_{A}B$, $g:={\kappa }_1$, and $h:={\kappa }_2$. Since ${\iota }_{A,B}$ is injective both maps ${\kappa }_1,{\kappa }_2:B\to B{+}_{A}B$ are injective too and, moreover, the pushout square is as well a pullback square. This ensures, especially, that ${\iota }_{A,B}:A\hookrightarrow B$ is the equalizer of the maps ${\kappa }_1,{\kappa }_2:B\to B{+}_{A}B$ in $\mathsf{Set}$. The pushout property of the square provides a unique map $k:B{+}_{A}B\to B$ such that

$${\kappa }_1;k={\kappa }_2;k=i{d}_B$$

(7)

Operations on C: We extend now C to a Σ-algebra $\mathcal{C}=(C,O{P}^{\mathcal{C}})$ by defining for each $\mathtt{op}$ in $OP$ a corresponding operation ${\mathtt{op}}^{\mathcal{C}}:{\left(B{+}_{A}B\right)}^{I_{\mathtt{op}}}\to {\left(B{+}_{A}B\right)}^{O_{\mathtt{op}}}$. For any $\mathbf{c}:I_{\mathtt{op}}\to B{+}_{A}B$ in ${\left(B{+}_{A}B\right)}^{I_{\mathtt{op}}}$ we do have four possible cases.

Case 1 

$\mathbf{c}$ factors through ${\kappa }_1$: There exists a map ${\mathit{b}}_1:I_{\mathtt{op}}\to B$ such that ${\mathit{b}}_1;{\kappa }_1=\mathbf{c}$ (see the right diagram above). ${\mathit{b}}_1$ is unique since ${\kappa }_1$ is a monomorphism. We simply set

$${\mathtt{op}}^{\mathcal{C}}\left(\mathbf{c}\right)={\mathtt{op}}^{\mathcal{C}}({\mathit{b}}_1;{\kappa }_1):={\mathtt{op}}^{\mathcal{B}}\left({\mathit{b}}_1\right);{\kappa }_1$$

(8)

Case 2 

$\mathbf{c}$ factors through ${\kappa }_2$: Analogously to Case 1.

Case 3 

Overlapping of Case 1 and 2: There exist maps ${\mathit{b}}_1,{\mathit{b}}_2:I_{\mathtt{op}}\to B$ such that ${\mathit{b}}_1;{\kappa }_1=\mathbf{c}={\mathit{b}}_2;{\kappa }_1$. Due to the pullback property of the square there exists a unique $\mathbf{a}:I_{\mathtt{op}}\to A$ such that ${\mathit{b}}_1=\mathbf{a};{\iota }_{A,B}={\mathit{b}}_2$. The homomorphism property of ${\iota }_{\mathcal{A},\mathcal{B}}$ ensures ${\mathtt{op}}^{\mathcal{B}}\left({\mathit{b}}_1\right)={\mathtt{op}}^{\mathcal{B}}(\mathbf{a};{\iota }_{A,B})={\mathtt{op}}^{\mathcal{B}}\left({\mathit{b}}_2\right)={\mathtt{op}}^{\mathcal{A}}\left(\mathbf{a}\right);{\iota }_{A,B}$ thus we get, finally, ${\mathtt{op}}^{\mathcal{B}}\left({\mathit{b}}_1\right);{\kappa }_1={\mathtt{op}}^{\mathcal{A}}\left(\mathbf{a}\right);{\iota }_{A,B};{\kappa }_1={\mathtt{op}}^{\mathcal{A}}\left(\mathbf{a}\right);{\iota }_{A,B};{\kappa }_2={\mathtt{op}}^{\mathcal{B}}\left({\mathit{b}}_2\right);{\kappa }_2$. That is, in the event of an overlapping, Case 1 and Case 2 define the same output ${\mathtt{op}}^{\mathcal{C}}\left(\mathbf{c}\right)={\mathtt{op}}^{\mathcal{B}}\left({\mathit{b}}_1\right);{\kappa }_1={\mathtt{op}}^{\mathcal{B}}\left({\mathit{b}}_2\right);{\kappa }_2$. Note, that there will always be an overlapping for constant symbols!

Case 4 

$\mathbf{c}$ factors neither through ${\kappa }_1$ nor through ${\kappa }_2$: This can only happen if $I_{\mathtt{op}}=I_n$ with $n\ge 2$. Utilizing the operations in $\mathcal{B}$ to a maximal extent, Cases 1 and 2 define a partial map from ${\left(B{+}_{A}B\right)}^{I_{\mathtt{op}}}$ to ${\left(B{+}_{A}B\right)}^{O_{\mathtt{op}}}$. However, since we restrict ourselves to total operations, we have to find an ad hoc totalization trick to turn ${\mathtt{op}}^{\mathcal{C}}$ into a total operation. Employing (7), we may decide to utilize the operations in $\mathcal{B}$ to produce outputs in the left copy of B in $B{+}_{A}B$. We set

$${\mathtt{op}}^{\mathcal{C}}\left(\mathbf{c}\right):={\mathtt{op}}^{\mathcal{B}}(\mathbf{c};k);{\kappa }_1$$

(9)

The operations in $\mathcal{C}$ are defined by (8) exactly in a way that the maps ${\kappa }_1$ and ${\kappa }_2$ become Σ-homomorphisms ${\kappa }_1,{\kappa }_2:\mathcal{B}\to \mathcal{C}$. Note, that Case 4 has no relevance for the homomorphism property of ${\kappa }_1$ and ${\kappa }_2$! Theorem 1 ensures, finally, that the inclusion Σ-homomorphism ${\iota }_{\mathcal{A},\mathcal{B}}:\mathcal{A}\to \mathcal{B}$ is the equalizer of the Σ-homomorphisms ${\kappa }_1,{\kappa }_2:\mathcal{B}\to \mathcal{C}$. □

Regularity of $\mathsf{Alg}\left(\mathsf{\Sigma }\right)$ entails that the concepts accessible and epic are equivalent.

Proposition 4

(Accessible ≅ Epic). For any Σ-homomorphism $f:\mathcal{A}\to \mathcal{B}$ it holds that $\mathcal{B}$ is accessible via the underlying map $f:A\to B$, i.e., in other words $\mathcal{B}$ is equal to the homomorphic image $\mathcal{R}(f,\mathcal{B})$ of $\mathcal{A}$ w.r.t. f, if, and only if, $f:\mathcal{A}\to \mathcal{B}$ is an epimorphism in $\mathsf{Alg}\left(\mathsf{\Sigma }\right)$.

Proof. 

“Accessible implies epic” has been shown in Lemma 1. We show now “epic implies accessible”: We consider an arbitrary Σ-subalgebra $\mathcal{X}$ of $\mathcal{B}$ such that there exists a map $f_{\mathcal{X}}:A\to X$ with $f_{\mathcal{X}};{\iota }_{X,B}=f$. Due to Proposition 3 there exist Σ-homomorphisms $g,h:\mathcal{B}\to \mathcal{C}$ such that ${\iota }_{\mathcal{X},\mathcal{B}}:\mathcal{X}\to \mathcal{B}$ is the equalizer of g and h. Due to the assumption $f_{\mathcal{X}};{\iota }_{X,B}=f$, we get $f;g=f;h$ and thus $g=h$ since f is epic. This means, however, $\mathcal{X}=\mathcal{B}$ and, finally, $\mathcal{R}(f,\mathcal{B})=\mathcal{B}$ due to the construction of $\mathcal{R}(f,\mathcal{B})$ in Definition 6. □

The axiom of choice is equivalent to the statement that all epimorphisms $f:A\to B$ in the category $\mathsf{Set}$ are split, i.e., there exists a map $g:B\to A$ such that $g;f=i{d}_B$. As a consequence each homomorphism between $\mathsf{\Sigma }$-algebras maps closed subsets to closed subsets. Especially, we have

Lemma 2

(Closed Images). For any Σ-homomorphism $f:\mathcal{A}\to \mathcal{B}$ the (set-theoretic) image $f\left(A\right)\subseteq B$ of the carrier A of $\mathcal{A}$ is closed in $\mathcal{B}$. We denote by $f\left(\mathcal{A}\right)$ the unique Σ-subalgebra of $\mathcal{B}$ with carrier $f\left(A\right)$ and call it the(set-theoretic) image of $\mathcal{A}$ w.r.t. the $\mathsf{\Sigma }$-homomorphism f.

Lemma 2 is the last brick we need to conclude that the epic $\mathsf{\Sigma }$-homomorphisms are exactly the surjective one.

Corollary 7

(Epic ≅ Surjective). For any Σ-homomorphism $f:\mathcal{A}\to \mathcal{B}$ we have $f\left(\mathcal{A}\right)=\mathcal{R}(f,\mathcal{B})$ thus $f:\mathcal{A}\to \mathcal{B}$ is an epimorphism in $\mathsf{Alg}\left(\mathsf{\Sigma }\right)$ if, and only if, the map $f:A\to B$ is surjective.

Proof. 

By Corollary 4 we have $f\left(A\right)\subseteq R(f,\mathcal{B})$. Lemma 2 gives us the Σ-subalgebra $f\left(\mathcal{A}\right)$ of $\mathcal{B}$ at hand and ensures $f\left(A\right)\supseteq R(f,\mathcal{B})$ due to the construction of $R(f,\mathcal{B})$ in Definition 6. This gives us $f\left(\mathcal{A}\right)=\mathcal{R}(f,\mathcal{B})$ since two Σ-subalgebras of a Σ-algebra $\mathcal{B}$ are equal if, and only if, they do have the same carrier. By Proposition 4 we get $f:\mathcal{A}\to \mathcal{B}$ epic if, and only if, $\mathcal{B}=f\left(\mathcal{A}\right)=\mathcal{R}(f,\mathcal{B})$. $\mathcal{B}=f\left(\mathcal{A}\right)$, however, means that f is surjective. □

Remark 5

(Stepwise Generation of Closed Subsets). There is another, more constructive, way to construct the closed sets $R(A,\mathcal{B})$. We start with A and add all the elements from B that we can reach by applying successively the operations in $\mathcal{B}$ to elements that have already been reached. A categorical analysis, formalization, and generalization of this stepwise iterative construction can be found in [12], for example.

3.3. Terms and Term Algebras

We define terms as strings of symbols. To distinguish terms from metalevel expressions, such as $o{p}^{\mathcal{A}}(a_1,\dots ,a_n)$, we will use angle bracket symbols $"\langle ","\rangle "$, instead of parenthesis $"(",")"$, to build terms. Moreover, we will use delimiter signs ⌜…⌝ to indicate that the expression between the delimiters is a string. So, the delimiter signs $⌜\dots ⌝$ are not constituents of terms and we may just drop them if convenient. The following is a traditional inductive definition of terms similar to [13,14]:

Definition 8

(Terms). The set $T_{\mathsf{\Sigma }}\left(X\right)$ of all $\mathsf{\Sigma }$-terms over a set X of variables is the smallest set of strings of symbols such that

Variables: 

$⌜x⌝\in T_{\mathsf{\Sigma }}\left(X\right)$, for all $x\in X$;

Constants: 

$⌜\mathtt{c}\langle \rangle ⌝\in T_{\mathsf{\Sigma }}\left(X\right)$, for all $\mathtt{c}\in OP$ with $I_{\mathtt{c}}=I_0$;

Operations: 

$⌜\mathtt{op}\langle t_1,\dots ,t_n\rangle ⌝\in T_{\mathsf{\Sigma }}\left(X\right)$, for all $\mathtt{op}\in OP$ with $I_{\mathtt{op}}=I_n$, $n\ge 1$ and all maps $\mathit{t}=(⌜t_1⌝,\dots ,⌜t_n⌝)$ in $T_{\mathsf{\Sigma }}{\left(X\right)}^{I_n}$.

Note that the assignments $x\mapsto ⌜x⌝$, assigning to each variable the string consisting only of a single symbol denoting this variable, define an injective map ${\eta }_X$: $X\to T_{\mathsf{\Sigma }}\left(X\right)$. Note, moreover, that in case $X=I_n=\{i_1,i_2,\dots ,i_n\}$ each operation symbol $\mathtt{op}\in OP$ is reborn as the $\mathsf{\Sigma }$-term $⌜\mathtt{op}\langle i_1,\dots ,i_n\rangle ⌝\in T_{\mathsf{\Sigma }}\left(I_n\right)$.

A term can be seen as a “tree-like computation scheme” and if we assign to variables certain values in an algebra we can compute a value in this algebra following this computation scheme. Terms are constructed inductively thus we can define this kind of evaluation of terms also inductively.

Definition 9

(Evaluation of terms). For any set X of variables, any Σ-algebra $\mathcal{A}=(A,O{P}^{\mathcal{A}})$ and any map $\alpha :X\to A$ (called a variable assignment ) we can define inductively a map ${\alpha }^{*}:T_{\mathsf{\Sigma }}\left(X\right)\to A$:

Variables: 

${\alpha }^{*}\left(x\right):=\alpha \left(x\right)$, for all $⌜x⌝\in T_{\mathsf{\Sigma }}\left(X\right)$;

Constants: 

${\alpha }^{*}\left(\mathtt{c}\langle \rangle \right):={\mathtt{c}}^{\mathcal{A}}\left(\right)\left(o\right)$, for all $⌜\mathtt{c}\langle \rangle ⌝\in T_{\mathsf{\Sigma }}\left(X\right)$;

Operations: 

${\alpha }^{*}\left(\mathtt{op}\langle t_1,\dots ,t_n\rangle \right):={\mathtt{op}}^{\mathcal{A}}({\alpha }^{*}\left(t_1\right),\dots ,{\alpha }^{*}\left(t_n\right))\left(o\right)$, for all $⌜\mathtt{op}\langle t_1,\dots ,t_n\rangle ⌝\in T_{\mathsf{\Sigma }}\left(X\right)$.

All three cases in Definition 9 are disjoint and terms are only equal if, and only if, they are equal as strings thus ${\alpha }^{*}$ is uniquely defined.

There is no indication in Definition 8 and Definition 9, respectively, where the sets $T_{\mathsf{\Sigma }}\left(X\right)$ of $\mathsf{\Sigma }$-terms live and where the evaluation of $\mathsf{\Sigma }$-terms takes place. A very common and powerful practice in Universal Algebra is to internalize $\mathsf{\Sigma }$-terms as elements of carriers of $\mathsf{\Sigma }$-algebras and to encode term evaluation by $\mathsf{\Sigma }$-homomorphisms: First, we observe that the stepwise construction of terms in Definition 8 can be reflected by defining for each operation symbol in $OP$ a corresponding (constructor) operation on $T_{\mathsf{\Sigma }}\left(X\right)$:

Definition 10

(Term algebra). For a set X of variables we define the term $\mathsf{\Sigma }$-algebra over X${\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)=(T_{\mathsf{\Sigma }}\left(X\right),O{P}^{{\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)})$ by

Constants: 

${\mathtt{c}}^{{\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)}\left(\right)\left(o\right):=⌜\mathtt{c}\langle \rangle ⌝$, for all $\mathtt{c}\in OP$ with $I_{\mathtt{c}}=I_0$, and

Operations: 

${\mathtt{op}}^{{\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)}\left(\mathit{t}\right)\left(o\right):=⌜\mathtt{op}\langle t_1,\dots ,t_n\rangle ⌝$, for all $\mathtt{op}\in OP$ with $I_{\mathtt{op}}=I_n$, $n\ge 1$ and all maps $\mathit{t}=(⌜t_1⌝,\dots ,⌜t_n⌝)$ in $T_{\mathsf{\Sigma }}{\left(X\right)}^{I_n}$.

Note that the term $\mathsf{\Sigma }$-algebra ${\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)$ is generated by ${\eta }_X\left(X\right)\subseteq T_{\mathsf{\Sigma }}\left(X\right)$. This is implicitly ensured by the statement of $T_{\mathsf{\Sigma }}\left(X\right)$ being the smallest set satisfying the conditions in Definition 8. We say that the elements in ${\eta }_X\left(X\right)$ are the generators of ${\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)$.

Second, we observe that the introduction of term $\mathsf{\Sigma }$-algebras ${\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)$ allows us to encode the defining equations for the cases Constants and Operations in Definition 9 by the requirement that the map ${\alpha }^{*}:T_{\mathsf{\Sigma }}\left(X\right)\to A$ should establish a $\mathsf{\Sigma }$-homomorphism ${\alpha }^{*}:{\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)\to \mathcal{A}$:

Constants: 

${\alpha }^{*}\left(\mathtt{c}\langle \rangle \right)={\alpha }^{*}\left({\mathtt{c}}^{{\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)}\left(\right)\left(o\right)\right)=({\mathtt{c}}^{{\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)}\left(\right);{\alpha }^{*})\left(o\right)={\mathtt{c}}^{\mathcal{A}}\left(\right)\left(o\right)$, for all $⌜\mathtt{c}\langle \rangle ⌝\in T_{\mathsf{\Sigma }}\left(X\right)$;

Operations: 

${\alpha }^{*}\left(\mathtt{op}\langle t_1,\dots ,t_n\rangle \right)={\alpha }^{*}\left({\mathtt{op}}^{{\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)}\left(\mathit{t}\right)\left(o\right)\right)=({\mathtt{op}}^{{\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)}\left(\mathit{t}\right);{\alpha }^{*})\left(o\right)={\mathtt{op}}^{\mathcal{A}}(\mathit{t};{\alpha }^{*})\left(o\right)$$={\mathtt{op}}^{\mathcal{A}}({\alpha }^{*}\left(t_1\right),\dots ,{\alpha }^{*}\left(t_n\right))\left(o\right)$, for all $⌜\mathtt{op}\langle t_1,\dots ,t_n\rangle ⌝\in T_{\mathsf{\Sigma }}\left(X\right)$ where $\mathit{t}$ is the map in $T_{\mathsf{\Sigma }}{\left(X\right)}^{I_n}$ defined by $\mathit{t}=(⌜t_1⌝,\dots ,⌜t_n⌝)$.

The third case Variables in Definition 9 simply requires that the map ${\alpha }^{*}:T_{\mathsf{\Sigma }}\left(X\right)\to A$ is an extension of the map $\alpha :X\to A$ thus the statement “Definition 9 defines ${\alpha }^{*}:T_{\mathsf{\Sigma }}\left(X\right)\to A$ uniquely” is transformed into the statement that the term $\mathsf{\Sigma }$-algebra ${\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)$ is a $\mathsf{\Sigma }$-algebra freely generated by X.

Proposition 5

(Term Algebras as Free Construction). Given a set X of variables, the Σ-term algebra ${\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)$ has the following universal property: For any Σ-algebra $\mathcal{A}=(A,O{P}^{\mathcal{A}})$ and any map $\alpha :X\to A$ there exists a unique Σ-homomorphism ${\alpha }^{*}:{\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)\to \mathcal{A}$ such that ${\eta }_X;{\alpha }^{*}=\alpha$.

The universal property in Proposition 5 characterizes ${\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)$ uniquely up to isomorphism and the case $X=\varnothing$ gives us initial $\mathsf{\Sigma }$-algebras at hand.

Corollary 8.

${\mathcal{T}}_{\mathsf{\Sigma }}(\varnothing )$ is initial in the category $\mathsf{Alg}\left(\mathsf{\Sigma }\right)$.

It is a standard result for free constructions that the assignments $X\mapsto {\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)$ and $(f:X\to Y)\mapsto ({(f;{\eta }_Y)}^{*}:{\mathcal{T}}_{\mathsf{\Sigma }}\left(X\right)\to {\mathcal{T}}_{\mathsf{\Sigma }}\left(Y\right))$ define a (free) functor ${\mathbf{F}}_{\mathsf{\Sigma }}:\mathsf{Set}\to \mathsf{Alg}\left(\mathsf{\Sigma }\right)$ that is left-adjoint to the forgetful functor ${\mathbf{U}}_{\mathsf{\Sigma }}$ (see [15]).