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Homesubalgebra of a partial algebra
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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 $B$ 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. 2. Let $A$ be the set of all nonnegative 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 $7A6=1\in B$, and and $(C,_{C})$ is not a subalgebra of $(A,_{A})$, since $1_{C}1$ is not defined in $C$, even though $1A1$ 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).
Mathematics Subject Classification
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