A set of finitary operations on a fixed set that contains all projection functions and is closed under composition (substitution of operations into arguments).

Principle

Closure under composition together with presence of projections characterizes the algebraic notion of all term operations definable from a given signature or algebra; clones capture the operational essence of an algebra's expressive power.

Demonstration

On a two-element set {0,1}, the set of all Boolean functions (of all finite arities) forms a clone; subsets of these closed under composition and containing projections correspond to classical clones in Post's lattice, such as the clone of monotone Boolean functions.

Misapplication

Calling any set of functions closed under pointwise combination a clone, or conflating clones with groups of permutations; such mistakes ignore the required projection functions and the compositional closure that define clones.

Consequence

Knowing a clone of operations determines which relations are preserved (polymorphisms) and thereby classifies definability and computational properties (for example, constraint satisfaction complexity depends on polymorphism clones).

Reversal

The dual perspective focuses on sets of relations closed under primitive positive definitions (co-clones); reversing the viewpoint from operations to relations yields complementary classification tools and different technical invariants.

Boundary

Applies only to finitary (finite-arity) operations on a single carrier set and requires all projections; excludes infinitary operations, multimodal outputs (multiple-output operations), and collections lacking projection functions.

Semantic Tension

Clone competes conceptually with monoid-of-operations or algebra generated by operations: a clone emphasizes closure under arbitrary composition and projections, whereas a generated algebra may refer to closure under specific primitive operations only, leading to different notions of definability.

Synthesis

A Clone is the full set of finitary term operations on a given set, closed under composition and containing projections; it encapsulates the operational power of an algebra and serves as the main invariant bridging operations and preserved relations.