# subalgebra of an algebraic system

Let $(A,O)$ be an algebraic system ($A\neq\varnothing$ is the underlying set and $O$ is the set of operators on $A$).

Let $B$ be a non-empty subset of $A$. $B$ is closed under operators of $A$ if for each $n$-ary operator $\omega_{A}$ on $A$, and any $b_{1},\ldots,b_{n}\in B$, we have $\omega_{A}(b_{1},\ldots,b_{n})\in B$.

Suppose $B$ is closed under operators of $A$. For each $n$-ary operator $\omega_{A}$ on $A$, we define $\omega_{B}:B^{n}\to B$ by $\omega_{B}(b_{1},\ldots,b_{n}):=\omega_{A}(b_{1},\ldots,b_{n})$. Each of these operators is well-defined and is called a restriction (of the corresponding $\omega_{A}$). Furthermore, $(B,O)$ is a well-defined algebraic system, and is called the subalgebra of $(A,O)$. When $(B,O)$ is a subalgebra of $(A,O)$, we also say that $(A,O)$ is an extension of $(B,O)$.

$(A,O)$ is clearly a subalgebra of itself. Any other subalgebra of $(A,O)$ is called a proper subalgebra.

Remark. If $(A,O)$ contains constants, then any subalgebra of $(A,O)$ must contain the exact same constants. For example, the ring $\mathbb{Z}$ of integers is an algebraic system with no proper subalgebras. Indeed, if $R$ is a subring of $\mathbb{Z}$, $1\in R$, so $R=\mathbb{Z}$.

Since we are operating under the same operator set, we can, for convenience, drop $O$ and simply call $A$ an algebra, $B$ a subalgebra of $A$, etc… If $B_{1},B_{2}$ are subalgebras of $A$, then $B_{1}\cap B_{2}$ is also a subalgebra. In fact, given any set of subalgebras $B_{i}$ of $A$, their intersection $\bigcap B_{i}$ is also a subalgebra.

Generating Set of an Algebra

Let $C$ be any subset of an algebra $A$. Consider the collection $[C]$ of all subalgebras of $A$ containing $C$. This collection is non-empty because $A\in[C]$. The intersection of all these subalgebras is again a subalgebra containing the set $C$. Denote this subalgebra by $\langle C\rangle$. $\langle C\rangle$ is called the subalgebra spanned by $C$, and $C$ is called the spanning set of $\langle C\rangle$. Conversely, any subalgebra $B$ of $A$ has a spanning set, namely itself: $B=\langle B\rangle$.

Given a subalgebra $B$ of $A$, a minimal spanning set $X$ of $B$ is called a generating set of $B$. By minimal we mean that the set obtained by deleting any element from $X$ no longer spans $B$. When $B$ has a generating set $X$, we also say that $X$ generates $B$. If $B$ can be generated by a finite set, we say that $B$ is finitely generated. If $B$ can be generated by a single element, we say that $B$ is cyclic.

Remark. $\langle\varnothing\rangle=$ the subalgebra generated by the constants of $A$. If no such constants exist, $\langle\varnothing\rangle:=\varnothing$.

From the discussion above, the set of subalgebras of an algebraic system forms a complete lattice. Given subalgebras $A_{i}$, $\bigvee A_{i}$ is the intersection of all $A_{i}$, and $\bigvee A_{i}$ is the subalgebra $\langle\bigcup A_{i}\rangle$. The lattice of all subalgebras of $A$ is called the subalgebra latttice of $A$, and is denoted by $\operatorname{Sub}(A)$.

 Title subalgebra of an algebraic system Canonical name SubalgebraOfAnAlgebraicSystem Date of creation 2013-03-22 16:44:19 Last modified on 2013-03-22 16:44:19 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 9 Author CWoo (3771) Entry type Definition Classification msc 08A30 Classification msc 08A05 Classification msc 08A62 Synonym subalgebra lattice Defines subalgebra Defines generating set Defines subalgebra generated by Defines extension of an algebraic system Defines restriction Defines proper subalgebra Defines lattice of subalgebras Defines spanning set Defines finitely generated Defines cyclic