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 $𝑨$ and $𝑩$ are partial algebras of type $\tau$:

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

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

2. $𝑩$ is a relative subalgebra of $𝑨$ if $B\subseteq A$, and $f_{𝑩}$ is a restriction of $f_{𝑨}$ relative to $B$ (http://planetmath.org/Subfunction) for every operator symbol $f\in \tau$.

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

3. $𝑩$ is a subalgebra of $𝑨$ if $B\subseteq A$, and $f_{𝑩}$ is a restriction (http://planetmath.org/Subfunction) of $f_{𝑨}$ for every operator symbol $f\in \tau$.

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

Notice that if $𝑩$ is a weak subalgebra of $𝑨$, then every constant of $𝑩$ is a constant of $𝑨$, 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 $\{0,1,\mathrm{\dots },n\}$ 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 $𝑩$ of $𝑨$ is a relative subalgebra iff given $b_1,\mathrm{\dots },b_n\in B$ such that $f_A(b_1,\mathrm{\dots },b_n)$ is defined and is in $B$, then $f_B(b_1,\mathrm{\dots },b_n)$ is defined. A relative subalgebra $𝑩$ of $𝑨$ is a subalgebra iff whenever $f_A(b_1,\mathrm{\dots },b_n)$ is defined for $b_i\in B$, it is in $B$.

2. Let $𝑨$ be a partial algebra of type $\tau$, and $B\subseteq A$. For each $n$-ary function symbol $f\in \tau$, define $f_{𝑩}$ on $B$ as follows: $f_{𝑩}(b_1,\mathrm{\dots },b_n)$ is defined in $B$ iff $f_{𝑨}(b_1,\mathrm{\dots },b_n)$ is defined in $A$ and $f_{𝑨}(b_1,\mathrm{\dots },b_n)\in B$. This turns $𝑩$ into a partial algebra. However, $𝑩$ may not be of type $\tau$, since $f_{𝑩}$ may not be defined at all on $B$. When $𝑩$ is a partial algebra of type $\tau$, it is a relative subalgebra of $𝑨$.

3. When $𝑨$ is an algebra, all three notions of subalgebras are equivalent (assuming that the partial operations on a weak subalgebra are all total).