Franklin's Notes

Algebraic system

An algebraic system consists of a set $\Omega$ of operators and a set $E$ of identities. The set $\Omega$ is graded, so that each operator $\omega\in\Omega$ has an arity. If $\omega$ has arity $1$, then it acts as a unary operator; if it has arity $2$, then it acts as a binary operator; and so on. We say that $\Omega$ acts on a set $S$ if each operator $\omega\in\Omega$ of arity $n$ is assigned to a function $\omega_A:S^n\to S$, and we say that the mapping $A$ which sends each $A:\omega\mapsto\omega_A$ is called the action of $\Omega$ on $S$. Of course, there may be many possible actions of a set of operators upon some set. If it's helpful to think of this in terms of model theory, we may think of $\Omega$ as the signature of a language in first-order logic with no relation symbols, and $S$ as the universe of a model of this language, with $A$ being the interpretation function.

From $\Omega$, we may also form a set $\Lambda$ of derived operators consisting of operators that can be formed in one of the following two ways:

1. Given $\omega$ of arity $n$ and operators $\lambda_1,\cdots,\lambda_n$ of arities $m_1,\cdots,m_n$, we may form the composition $\omega(\lambda_1,\cdots,\lambda_n)$.
2. Given a function $f$ from ${1,\cdots,n}$ to ${1,\cdots,m}$ and a derived operator $\lambda$ of arity $n$, we can form a derived operator described in terms of variables as $\theta(x_1,\cdots,x_m)=\lambda(x_{f1},\cdots,x_{fn})$.

The set of identities $E$ of the algebraic system consists of ordered pairs $\langle \lambda,\mu\rangle$ of derived operators, and an action $A$ of $\Omega$ on a set $A$ is said to satisfy $\langle \lambda,\mu\rangle$ if $\lambda_A=\mu_A$ as functions $S^n\to S$. A $\langle \Omega,E\rangle$-algebra is a set $S$ together with an action $A$ of $\Omega$ on $S$ satisfying all of the identities of $E$.

Given a fixed algebraic system $\langle \Omega,E\rangle$, we can assemble all of its algebras into a category $\langle \Omega,E\rangle\text -\mathsf{Alg}$ called the category of $\langle\Omega,E\rangle$-algebras. The (homo)morphisms $g:A\to A'$ of $\langle \Omega,E\rangle$-algebras are defined as functions $g:S\to S'$ on the underlying sets with the property that for all $\omega\in\Omega$ and $x_1,\cdots,x_n\in S$.




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