Universal Algebra/Definitions, examples

Recall that whenever $S$ is a set, then ${\mathcal {T}}(S)$ is the class of all $S$-tuples, regardless of size.

Definition (operation):

Let $A$ be a set. An operation on $A$ is a subclass $\alpha \subseteq {\mathcal {T}}(A)$ together with a function $\circ :\alpha \to A$.

Note that in this definition, $\alpha =\{()\}$ (the set consisting only of the empty tuple) is allowed, so that in this case, $\circ$ can be identified with a constant in $A$. It is customary to regard $0$-ary operations as constants in $A$.

Definition (arity):

Let $A$ be a set together with an operation $\circ$. If $c$ is a cardinal, and $\circ$ is defined precisely on tuples of cardinality $c$, then $c$ is called the arity of the operation $\circ$. $\circ$ is then called $c$-ary.

Definition (arbitr-ary):

Let $A$ be a set together with an operation $\circ$. If $\circ$ is defined on all of ${\mathcal {T}}(A)$, it is called arbitr-ary.

Proposition (intersections preserve closure properties):

Let $S$ be an instance of an algebraic variety with operations $\circ _{1},\ldots ,\circ _{n}$, and let $(M_{\alpha })_{\alpha \in A}$ be a family of subsets of $S$ such that each $M_{\alpha }$ is closed under all operations $\circ _{1},\ldots ,\circ _{n}$. Then $\cap _{\alpha \in A}M_{\alpha }$ is also closed under the operations $\circ _{1},\ldots ,\circ _{n}$.

Proof: Suppose that $j\in [n]$ and in order to ease notation, define $M:=\cap _{\alpha \in A}M_{\alpha }$. Suppose that $\sigma \in {\mathcal {T}}(M)$ is a tuple that lies in the domain of definition of $\circ _{j}$. Then by assumption, for each $\alpha \in A$ $\circ _{j}(\sigma )$ lies in $M_{\alpha }$, so that $\circ _{j}(\sigma )$ lies in $M$. $\Box$

Definition (algebraic variety):

An algebraic variety is the class of all sets $A$ with certain operations $(\circ _{\beta })_{\alpha \in B}$ (where $B$ is a fixed index set, specific to the algebraic variety at hand), so that for each $\beta$ the operation $\circ _{\beta }$ is defined on all tuples which are defined by a set-theoretic expression that only depends on $\beta$ and the other operations, and so that a set $(S_{\gamma })_{\gamma \in \Gamma }$ of rules hold for the operations, where a rule is defined as follows:

1. A term is recursively defined as follows:
   1. All $0$-ary operations (and tuples thereof) are terms
   2. All variables (which for our purposes are just letters) are terms
   3. Whenever $(t_{\delta })_{\delta \in \Delta }$ is a tuple of terms, $\circ _{\beta }((t_{\delta })_{\delta \in \Delta })$ is a term
2. A rule is then an expression of the form $t=s$, where $s$ and $t$ are terms
3. The rule is said to hold for the given algebraic structure iff the identity given by it holds whenever the variables are each replaced by adequate tuples, so that the expressions of the term all make sense (ie. the operations of $A$ are defined on all resulting tuples)

Definition (algebraic structure):

An algebraic structure of a given algebraic variety is an element of the given algebraic variety.

Definition (substructure):

If $A$ is an algebraic structure of a given algebraic variety and $B\subseteq A$ is a subset which, equipped with the restrictions of the operations of $A$ is itself an algebraic structure of that algebraic variety, $B$ is called a substructure of $A$.

Proposition (closedness under operations means algebraic structure):

Let $A$ be an algebraic structure, and let $B\subseteq A$ be a subset that is closed under all the operations that go along with $A$. Then $B$ is an algebraic structure of the same algebraic variety as $A$.

For example, if we have a subset of a group that contains the identity and is closed under inversion and the product (that is, if we have a subset of a group that is closed under the 0-ary, the 1-ary and the 2-ary operation), then that subset is a subgroup.

Proof: It suffices to note that the validity of the rules is not infringed, since all we do is to quantify over a smaller class. $\Box$

Proposition (greatest lower bound structure is intersection):

Let $A$ be a set together with operations $(\circ _{\alpha })_{\alpha \in A}$ on it, and let $(B_{\gamma })_{\gamma \in \Gamma }$ be a nontrivial family of subsets of $A$ that are all algebraic structures of the same algebraic variety, when the operations of $A$ are restricted to them. Then their greatest lower bound algebraic structure with respect to set inclusion is given by the intersection $\cap _{\gamma \in \Gamma }B_{\gamma }$.

Proof: The intersection $\cap _{\gamma \in \Gamma }B_{\gamma }$ is closed under all intersections since intersections preserve closure properties, and hence, since a subset that is closed under the operations is a substructure itself, it is a substructure of any of the $B_{\alpha }$, that is, itself an algebraic structure. Clearly, it is the largest that is contained all $B_{\gamma }$, since it consists exactly of the elements that are contained in all $B_{\gamma }$, and hence any additional element would not be countained in all $B_{\gamma }$. $\Box$