General algebraic systems Varieties Example

Trivial variety

Builds on Identity Lattice of subvarieties

Example

Fix a signature $\sigma$ . The trivial variety of signature $\sigma$ is the class of all algebras of signature $\sigma$ that satisfy the identity $x = y$ . An algebra satisfies $x = y$ exactly when any two of its elements are equal, so the trivial variety consists of the one-element algebras. It is a variety because it is defined by an identity. The check by closure is also direct. A subalgebra or a homomorphic image of a one-element algebra has one element. A direct product of one-element algebras has one element, and the empty product is a one-element algebra too.

Remarks

Every variety of signature $\sigma$ contains all one-element algebras of signature $\sigma$ , since a one-element algebra satisfies every identity. So the trivial variety is the least element of the lattice of subvarieties of any variety. Its equational theory is the largest one: it contains every identity of signature $\sigma$ .

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`ES6P7WHJG9`
URI	`m2://ES6P7WHJG9`
Kind	example
Writing	W0 stub
MSC	08B15
Instance of	Variety of algebras
Uses	Identity, Lattice of subvarieties
Changed	2 November 2026

Trivial variety

Example

Remarks

Reading path

Further reading

What this page needs for W1

What links here