# subalgebra of an algebraic system

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

Subalgebras^{} of an Algebra^{}

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 ${b}_{1},\mathrm{\dots},{b}_{n}\in B$, we have ${\omega}_{A}({b}_{1},\mathrm{\dots},{b}_{n})\in B$.

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 ${\omega}_{B}({b}_{1},\mathrm{\dots},{b}_{n}):={\omega}_{A}({b}_{1},\mathrm{\dots},{b}_{n})$. 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 $\u27e8C\u27e9$. $\u27e8C\u27e9$ is called the subalgebra *spanned* by $C$, and $C$ is called the *spanning set* of $\u27e8C\u27e9$. Conversely, any subalgebra $B$ of $A$ has a spanning set, namely itself: $B=\u27e8B\u27e9$.

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. $\u27e8\mathrm{\varnothing}\u27e9=$ the subalgebra generated by the constants of $A$. If no such constants exist, $\u27e8\mathrm{\varnothing}\u27e9:=\mathrm{\varnothing}$.

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 $\u27e8\bigcup {A}_{i}\u27e9$. The lattice^{} of all subalgebras of $A$ is called the *subalgebra latttice* of $A$, and is denoted by $\mathrm{Sub}(A)$.

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 |