Polynomials in an Algebraic System

1. the projection $\pi_i:A^n\to A$ is an element of $S$ for each $i$ between $1$ and $n$ .
2. for every $0$ -ary operator symbol $c\in O$ , the corresponding element $c_A\in A$ is an element of $S$ .
3. for any $m$ -ary operator $\omega_A$ on $A$ , and any $p_1,\ldots,p_m\in S$ , the function $\omega_A(p_1,\ldots,p_m):A^n\to A$ , given by $$\omega_A(p_1,\ldots,p_m)(a_1,\ldots,a_n):=\omega_A(p_1(a_1,\ldots,a_n),\ldots, p_m(a_1,\ldots,a_n))$$ is an element of $S$ .

Examples.

• Let $G$ be a group. Then $f(x)=x$ is a unary polynomial. In general, a unary polynomial of $G$ has the form $x^n$ , $n\ge 1$ . For a binary polynomial, first note that $f(x,y)=x$ and $g(x,y)=y$ are both binary. For this, a binary polynomial of $G$ looks like $$x^{n_1}y^{m_1}\cdots x^{n_k}y^{m_k},$$ where $n_1,m_k\ge 0$ and the rest are positive integers. If $G$ is abelian, a binary polynomial has the simplified form $x^ny^m$ , and more generally, an $n$ -ary polynomial of an abelian group has the form $$x_1^{n_1}\cdots x_k^{n_k}.$$
• Let $R$ be a ring. $f(x)=x$ is a unary polynomial, so are $g(x)=nx$ and $h(x)=x^m$ where $n,m$ are non-negative integers. In general, a unary polynomial of $R$ looks like $$n_mx^{m}+n_{m-1}x^{m-1}+\cdots + n_1x+n_0,$$ where $n_i$ are non-negative integers with $n_m>0$ , and $m$ is any positive integer. This is the form of a polynomial that is familiar to most people.
• Continuing from the example above, note that, however, $$a_mx^{m}+a_{m-1}x^{m-1}+\cdots + a_1x+a_0$$ where $a_i\in R$ is in general not a polynomial under this definition. This is an example of an algebraic function in an algebraic system.

Remarks.

• ${P}_0(A)$ is non-empty iff $O$ contains constant symbols (nullary function symbols).
• By virtue of the definition, ${P}_n(A)$ is an algebraic system for each non-negative integer $n$ .

Polynomial Symbols

1. $x_1,\ldots,x_n$ ,
2. every nullary operator symbol,
3. if $\omega\in O$ is $m$ -ary and $p_1,\ldots,p_m\in S$ , then $\omega(p_1, \ldots, p_m)\in S$ .

Now, suppose $A\in K$ . For a given non-negative integer $n$ , consider the function $f_A:{P}_n(O)\to {P}_n(A)$ , defined recursively by

1. $f_A(x_i):=\pi_i$ ,
2. $f_A(c):=c_A$ ,
3. $f_A(\omega(p_1,\ldots,p_n)):=\omega_A(f_A(p_1),\ldots,f_A(p_n))$ .

Example. $x\vee(y\wedge z)$ and $(x\vee y)\wedge (x\vee z)$ are both polynomials symbols in the class of lattices. In the subclass of distributive lattices, the induced polynomials are equivalent. However, in the subclass of modular lattices, the equivalence no longer holds.

Remark. Like ${P}_n(A)$ , ${P}_n(O)$ is also an algebraic system, called the $n$ -ary polynomial algebra, or absolutely free algebra, of $O$ . Furthermore, the function $f_A$ above is an algebra homomorphism.

Bibliography

1

G. Grätzer: Universal Algebra, 2nd Edition, Springer, New York (1978).

2

S. Burris, H.P. Sankappanavar: A Course in Universal Algebra, Springer, New York (1981).