1. the definition of an operation, $f\colon A^n\to A$,
2. the definition of the arity of an operation, $\alpha(f)\in \mathbb{N}$,
3. the fact that there are operations of arity $0$,
4. the definition of a signature, $\sigma=(F,\alpha)$,
5. the definition of an algebra of signature $\sigma$, $\mathbb{A}=\langle A; F^{\mathbb{A}}\rangle$,
6. the definition of the universe, $A$, of an algebra $\mathbb{A}=\langle A; F^{\mathbb{A}}\rangle$.

• $\langle \mathbb{N}; S(x), 0\rangle$,
• $\langle \mathbb{N}; +, \cdot, S(x), 0\rangle$,
• $\mathbb{B}_2=\langle \{0,1\}; \wedge,\vee,\neg,0,1\rangle$.

1. Let $A$ be an army and for $x,y\in A$ let $\diamondsuit(x,y)$ = the least common commander of $x$ and $y$. $\diamondsuit$ is an operation on $A$, hence $\langle A; \diamondsuit\rangle$ is an algebra. (This kind of algebra can be used to model any hierarchical structure where any two elements have a least common upper bound.)
2. Any scheme for merging ranked lists of voter preferences is an operation on the set of ranked lists.
3. Procedures for combining data structures (insertion, deletion, merging) are operations on the class of data structures.

We let $S$ be a set and $\mathcal{F}=\{f\;|\;f\colon S\to S\}$ be the set of functions from $S$ to $S$. Let $1=\textrm{id}_S$ be the identity function from $S$ to $S$. We argued that the function algebra, $\langle \mathcal{F}; \circ, 1\rangle$, is a structure with an associative multiplication $\circ$ and a unit element $1$. (The associative law is $(\forall f)(\forall g)(\forall h)(f\circ (g\circ h)=(f\circ g)\circ h)$) and the unit laws are $(\forall f)(1\circ f=f)$ and $(\forall f)(f\circ 1=f)$). We defined the class of monoids to be the class of all algebraic structures $\langle M; \circ, 1\rangle$ that have an associative multiplication and a unit element. Monoids will turn out to be the correct algebraic model for functions under composition. We will be able to use monoids to prove that the universally quantified laws of functional composition which involve the identity function are exactly the consequences of the associative law and the unit laws.

We reviewed this handout on functions, and then introduced some function-related terminology for algebra, namely:

1. homomorphism,
2. embedding,
3. isomorphism
4. subuniverse/subalgebra

We discussed these slides, which prove the Cayley Representation Theorem for monoids.
Quiz 1.

1. A semigroup is an algebra $\langle S; \circ\rangle$ with a single associative binary operation. Semigroups is the “correct” algebraization of functions under composition, $\langle \textrm{Funct}(X,X); \circ\rangle$.
2. A left cancellative semigroup is a semigroup $\langle S; \circ\rangle$ with a single associative binary operation which also satisfies \[(\forall f)(\forall g)(\forall h)((f\circ g=f\circ h)\to (g=h)). \] Left cancellative semigroups are the “correct” algebraization of injective functions under composition.
3. A right cancellative semigroup is a semigroup $\langle S; \circ\rangle$ with a single associative binary operation which also satisfies \[(\forall f)(\forall g)(\forall h)((f\circ h=g\circ h)\to (f=g)). \] Right cancellative semigroups are the “correct” algebraization of surjective functions under composition.
4. A group is an algebraic structure $\langle G; \circ, {}^{-1},1\rangle$ with a binary operation $x\circ y$ for which the associative law holds, a unary operation $x^{-1}$ for which the inverse laws hold, and a constant $1$ for which the identity laws hold. Groups are the “correct” algebraization of bijective functions under composition.

We discussed examples of groups.

Quiz 2.

The group of all permutations of a set $X$ is called the symmetric group on $X$. We examined the Cayley Representation for monoids and found that it extends to a representation for groups. This shows that every group is embeddable in a symmetric group.

We tried to determine all small groups up to isomorphism, but we did not get far. We only showed that there is 1 group of size 1 (up to isomorphism) and 1 group of size 2 (up to isomorphism).