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Difference between revisions of "Free algebra"

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''in a class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f0414901.png" /> of universal algebras''
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''in a class $\mathfrak K$ of universal algebras''
  
An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f0414902.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f0414903.png" /> with a free generating system (or base) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f0414904.png" />, that is, with a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f0414905.png" /> of generators such that every mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f0414906.png" /> into any algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f0414907.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f0414908.png" /> can be extended to a homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f0414909.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f04149010.png" /> (see [[Free algebraic system|Free algebraic system]]). Any non-empty class of algebras that is closed under subalgebras and direct products and that contains non-singleton algebras, has free algebras. In particular, free algebras always exist in non-trivial varieties and quasi-varieties of universal algebras (see [[Variety of universal algebras|Variety of universal algebras]]; [[Algebraic systems, quasi-variety of|Algebraic systems, quasi-variety of]]). A free algebra in the class of all algebras of a given signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f04149011.png" /> is called absolutely free. An algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f04149012.png" /> of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f04149013.png" /> is a free algebra in some class of universal algebras of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f04149014.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f04149015.png" /> is intrinsically free, that is, if it has a generating set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f04149016.png" /> such that every mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f04149017.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f04149018.png" /> can be extended to an endomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f04149019.png" />. If a free algebra has an infinite base, then all its bases have the same cardinality (see [[Free Abelian group|Free Abelian group]]; [[Free algebra over a ring|Free algebra over a ring]]; [[Free associative algebra|Free associative algebra]]; [[Free Boolean algebra|Free Boolean algebra]]; [[Free group|Free group]]; [[Free semi-group|Free semi-group]]; [[Free lattice|Free lattice]]; [[Free groupoid|Free groupoid]]; [[Free module|Free module]]; and also [[Free product|Free product]]). Clearly, every element of a free algebra with a base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f04149020.png" /> can be written as a word over the alphabet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041490/f04149021.png" /> in the signature of the class being considered. It is natural to ask: When are different words equal as elements of the free algebra? In certain cases the answer is almost trivial (semi-groups, rings, groups, associative algebras), while in others it is fairly complicated (Lie algebras, lattices, Boolean algebras), and sometimes it does not have a recursive solution (alternative rings).
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An algebra $F$ in $\mathfrak K$ with a free generating system (or base) $X$, that is, with a set $X$ of generators such that every mapping of $X$ into any algebra $A$ from $\mathfrak K$ can be extended to a homomorphism of $F$ into $A$ (see [[Free algebraic system|Free algebraic system]]). Any non-empty class of algebras that is closed under subalgebras and direct products and that contains non-singleton algebras, has free algebras. In particular, free algebras always exist in non-trivial varieties and quasi-varieties of universal algebras (see [[Variety of universal algebras|Variety of universal algebras]]; [[Algebraic systems, quasi-variety of|Algebraic systems, quasi-variety of]]). A free algebra in the class of all algebras of a given signature $\Lambda$ is called absolutely free. An algebra $A$ of signature $\Lambda$ is a free algebra in some class of universal algebras of signature $\Lambda$ if and only if $A$ is intrinsically free, that is, if it has a generating set $X$ such that every mapping of $X$ into $A$ can be extended to an endomorphism of $A$. If a free algebra has an infinite base, then all its bases have the same cardinality (see [[Free Abelian group|Free Abelian group]]; [[Free algebra over a ring|Free algebra over a ring]]; [[Free associative algebra|Free associative algebra]]; [[Free Boolean algebra|Free Boolean algebra]]; [[Free group|Free group]]; [[Free semi-group|Free semi-group]]; [[Free lattice|Free lattice]]; [[Free groupoid|Free groupoid]]; [[Free module|Free module]]; and also [[Free product|Free product]]). Clearly, every element of a free algebra with a base $X$ can be written as a word over the alphabet $X$ in the signature of the class being considered. It is natural to ask: When are different words equal as elements of the free algebra? In certain cases the answer is almost trivial (semi-groups, rings, groups, associative algebras), while in others it is fairly complicated (Lie algebras, lattices, Boolean algebras), and sometimes it does not have a recursive solution (alternative rings).
  
  

Latest revision as of 13:01, 9 April 2014

in a class $\mathfrak K$ of universal algebras

An algebra $F$ in $\mathfrak K$ with a free generating system (or base) $X$, that is, with a set $X$ of generators such that every mapping of $X$ into any algebra $A$ from $\mathfrak K$ can be extended to a homomorphism of $F$ into $A$ (see Free algebraic system). Any non-empty class of algebras that is closed under subalgebras and direct products and that contains non-singleton algebras, has free algebras. In particular, free algebras always exist in non-trivial varieties and quasi-varieties of universal algebras (see Variety of universal algebras; Algebraic systems, quasi-variety of). A free algebra in the class of all algebras of a given signature $\Lambda$ is called absolutely free. An algebra $A$ of signature $\Lambda$ is a free algebra in some class of universal algebras of signature $\Lambda$ if and only if $A$ is intrinsically free, that is, if it has a generating set $X$ such that every mapping of $X$ into $A$ can be extended to an endomorphism of $A$. If a free algebra has an infinite base, then all its bases have the same cardinality (see Free Abelian group; Free algebra over a ring; Free associative algebra; Free Boolean algebra; Free group; Free semi-group; Free lattice; Free groupoid; Free module; and also Free product). Clearly, every element of a free algebra with a base $X$ can be written as a word over the alphabet $X$ in the signature of the class being considered. It is natural to ask: When are different words equal as elements of the free algebra? In certain cases the answer is almost trivial (semi-groups, rings, groups, associative algebras), while in others it is fairly complicated (Lie algebras, lattices, Boolean algebras), and sometimes it does not have a recursive solution (alternative rings).


Comments

Sometimes the phrases "free algebra" and "free algebraic system" are identified in meaning. See Free algebraic system.

How to Cite This Entry:
Free algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_algebra&oldid=12553
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article