Unlike an algebraic system, where there is only one way to define a subalgebra, there are several ways to define a subalgebra of a partial algebra.

Suppose $\boldsymbol{A}$ and $\boldsymbol{B}$ are partial algebras of type $\tau$ :

1. $\boldsymbol{B}$ is a weak subalgebra of $\boldsymbol{A}$ if $B\subseteq A$ , and $f_{\boldsymbol{B}}$ is a subfunction of $f_{\boldsymbol{A}}$ for every operator symbol $f\in \tau$ .

   In words, $\boldsymbol{B}$ is a weak subalgebra of $\boldsymbol{A}$ iff $B\subseteq A$ , and for each $n$ -ary symbol $f\in \tau$ , if $b_1,\ldots, b_n \in B$ such that $f_B(b_1,\ldots, b_n)$ is defined, then $f_A(b_1,\ldots, b_n)$ is also defined, and is equal to $f_B(b_1, \ldots, b_n)$ .

2. $\boldsymbol{B}$ is a relative subalgebra of $\boldsymbol{A}$ if $B\subseteq A$ , and $f_{\boldsymbol{B}}$ is a restriction of $f_{\boldsymbol{A}}$ relative to $B$ for every operator symbol $f\in \tau$ .

   In words, $\boldsymbol{B}$ is a relative subalgebra of $\boldsymbol{A}$ iff $B\subseteq A$ , and for each $n$ -ary symbol $f\in \tau$ , given $b_1,\ldots, b_n \in B$ , $f_B(b_1,\ldots, b_n)$ is defined iff $f_A(b_1,\ldots, b_n)$ is and belongs to $B$ , and they are equal.

3. $\boldsymbol{B}$ is a subalgebra of $\boldsymbol{A}$ if $B\subseteq A$ , and $f_{\boldsymbol{B}}$ is a restriction of $f_{\boldsymbol{A}}$ for every operator symbol $f\in \tau$ .

   In words, $\boldsymbol{B}$ is a subalgebra of $\boldsymbol{A}$ iff $B\subseteq A$ , and for each $n$ -ary symbol $f\in \tau$ , given $b_1,\ldots, b_n \in B$ , $f_B(b_1,\ldots, b_n)$ is defined iff $f_A(b_1,\ldots, b_n)$ is, and they are equal.

Notice that if $\boldsymbol{B}$ is a weak subalgebra of $\boldsymbol{A}$ , then every constant of $\boldsymbol{B}$ is a constant of $\boldsymbol{A}$ , and vice versa.

Every subalgebra is a relative subalgebra, and every relative subalgebra is a weak subalgebra. But the converse is false for both statements. Below are two examples.

1. Let $F$ be a field. Then every subalgebra of $F$ is a subfield, and every relative subalgebra of $F$ is a subring.
2. Let $A$ be the set of all non-negative integers, and $-_A$ the ordinary subtraction on integers. Consider the partial algebra $(A, -_A)$ .
   • Let $B=A$ and $-_B$ the usual subtraction on integers, but $x-_B y$ is only defined when $x,y\in B$ have the same parity. Then $(B, -_B)$ is a weak subalgebra of $(A, -_A)$ .
   • Let $C$ be the set of all positive integers, and $-_C$ the ordinary subtraction. Then $(C,-_C)$ is a relative subalgebra of $(A,-_A)$ .
   • Let $D$ be the set $\lbrace 0,1,\ldots, n\rbrace$ and $-_D$ the ordinary subtraction. Then $(D,-_D)$ is a subalgebra of $(A,-_A)$ .
   Notice that $(B,-_B)$ is not a relative subalgebra of $(A,-_A)$ , since $7-_B 6$ is not defined, even though $7 -A 6 = 1\in B$ , and and $(C,-_C)$ is not a subalgebra of $(A,-_A)$ , since $1 -_C 1$ is not defined in $C$ , even though $1 -A 1$ is defined in $A$ .

Remarks.

1. A weak subalgebra $\boldsymbol{B}$ of $\boldsymbol{A}$ is a relative subalgebra iff given $b_1,\ldots, b_n\in B$ such that $f_A(b_1,\ldots, b_n)$ is defined and is in $B$ , then $f_B(b_1,\ldots, b_n)$ is defined. A relative subalgebra $\boldsymbol{B}$ of $\boldsymbol{A}$ is a subalgebra iff whenever $f_A(b_1,\ldots, b_n)$ is defined for $b_i\in B$ , it is in $B$ .
2. Let $\boldsymbol{A}$ be a partial algebra of type $\tau$ , and $B\subseteq A$ . For each $n$ -ary function symbol $f\in \tau$ , define $f_{\boldsymbol{B}}$ on $B$ as follows: $f_{\boldsymbol{B}}(b_1,\ldots, b_n)$ is defined in $B$ iff $f_{\boldsymbol{A}}(b_1,\ldots, b_n)$ is defined in $A$ and $f_{\boldsymbol{A}}(b_1,\ldots, b_n)\in B$ . This turns $\boldsymbol{B}$ into a partial algebra. However, $\boldsymbol{B}$ may not be of type $\tau$ , since $f_{\boldsymbol{B}}$ may not be defined at all on $B$ . When $\boldsymbol{B}$ is a partial algebra of type $\tau$ , it is a relative subalgebra of $\boldsymbol{A}$ .
3. When $\boldsymbol{A}$ is an algebra, all three notions of subalgebras are equivalent (assuming that the partial operations on a weak subalgebra are all total).

Bibliography

1

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