##  [Clone](/clone-0) 

 Definition

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

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

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

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

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

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

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

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

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.