# subalgebra of a partial algebra

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. 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. 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 $B$ (http://planetmath.org/Subfunction) 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. 3.

$\boldsymbol{B}$ is a subalgebra of $\boldsymbol{A}$ if $B\subseteq A$, and $f_{\boldsymbol{B}}$ is a restriction (http://planetmath.org/Subfunction) 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. 1.

Let $F$ be a field. Then every subalgebra of $F$ is a subfield, and every relative subalgebra of $F$ is a subring.

2. 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 $\{0,1,\ldots,n\}$ 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 $7-A6=1\in B$, and and $(C,-_{C})$ is not a subalgebra of $(A,-_{A})$, since $1-_{C}1$ is not defined in $C$, even though $1-A1$ is defined in $A$.

Remarks.

1. 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. 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. 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).

## References

• 1 G. Grätzer: , 2nd Edition, Springer, New York (1978).
Title subalgebra of a partial algebra SubalgebraOfAPartialAlgebra 2013-03-22 18:42:54 2013-03-22 18:42:54 CWoo (3771) CWoo (3771) 10 CWoo (3771) Definition msc 08A55 msc 03E99 msc 08A62 weak subalgebra relative subalgebra subalgebra