Editing Universal algebra (section)

== Basic constructions ==
We assume that the type, <math>\Omega</math>, has been fixed.  Then there are three basic constructions in universal algebra: homomorphic image, subalgebra, and product.

A [[homomorphism]] between two algebras ''A'' and ''B'' is a [[function (mathematics)|function]] {{nowrap|''h'' : ''A'' → ''B''}} from the set ''A'' to the set ''B'' such that, for every operation ''f''<sub>''A''</sub> of ''A'' and corresponding ''f''<sub>''B''</sub> of ''B'' (of arity, say, ''n''), ''h''(''f''<sub>''A''</sub>(''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>))&nbsp;= ''f''<sub>''B''</sub>(''h''(''x''<sub>1</sub>), ..., ''h''(''x''<sub>''n''</sub>)).  (Sometimes the subscripts on ''f'' are taken off when it is clear from context which algebra the function is from.) For example, if ''e'' is a constant (nullary operation), then ''h''(''e''<sub>''A''</sub>)&nbsp;=&nbsp;''e''<sub>''B''</sub>.  If ~ is a unary operation, then ''h''(~''x'')&nbsp;= ~''h''(''x'').  If ∗ is a binary operation, then ''h''(''x''&nbsp;∗&nbsp;''y'')&nbsp;= ''h''(''x'')&nbsp;∗&nbsp;''h''(''y'').  And so on.  A few of the things that can be done with homomorphisms, as well as definitions of certain special kinds of homomorphisms, are listed under ''[[Homomorphism]]''.  In particular, we can take the homomorphic image of an algebra, ''h''(''A'').

A subalgebra of ''A'' is a subset of ''A'' that is closed under all the operations of ''A''.  A product of some set of algebraic structures is the [[cartesian product]] of the sets with the operations defined coordinatewise.