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.

$\boldsymbol{B}$ 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 $B$ (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.

$\boldsymbol{B}$ 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.

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 $\{0,1,\ldots,n\}$ 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.

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).

References

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

Title	subalgebra of a partial algebra
Canonical name	SubalgebraOfAPartialAlgebra
Date of creation	2013-03-22 18:42:54
Last modified on	2013-03-22 18:42:54
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	10
Author	CWoo (3771)
Entry type	Definition
Classification	msc 08A55
Classification	msc 03E99
Classification	msc 08A62
Defines	weak subalgebra
Defines	relative subalgebra
Defines	subalgebra