nLab trivial algebra

Definition

Every algebraic category (in just about any sense of the term) has a terminal object which is usually called the trivial algebra, often denoted $0$ or $1$. Its underlying set is a singleton, and every algebraic operation $1^n \to 1$ is the unique function to the singleton set.

Examples

• The singleton set is itself the trivial set.
• The trivial monoid is the trivial monoid.
• The trivial group is the trivial group.
• The trivial ring is the trivial ring.
• The $0$-dimensional vector space is the trivial vector space.
• The trivial Lie algebra is the trivial Lie algebra.
• The trivial category is the trivial strict category.