In universal algebra, a clone is a set $C$ of finitary operations on a fixed non‑empty set $A$ that satisfies two closure conditions:
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Containment of projections – for each arity $n\ge 1$ and each $i\in{1,\dots ,n}$, the $i$-th projection map
$$ \pi_i^n(x_1,\dots ,x_n)=x_i $$ belongs to $C$. -
Closure under composition – if $f\in C$ is an $m$-ary operation and $g_1,\dots ,g_m\in C$ are $n$-ary operations, then the composed operation
$$ h(x_1,\dots ,x_n)=f\bigl(g_1(x_1,\dots ,x_n),\dots ,g_m(x_1,\dots ,x_n)\bigr) $$ also belongs to $C$.
A clone therefore consists of all operations that can be built from a given collection of basic operations by repeatedly applying composition, while always retaining the projection operations. The notion abstracts the set of term operations of an algebraic structure: for any algebra $\mathbf{A}=(A,{f_i}_{i\in I})$, the set of all term operations on $A$ (i.e., operations definable by algebraic terms using the basic operations $f_i$) forms a clone, called the term clone of $\mathbf{A}$.
Relationship to Other Concepts
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Varieties and equational theories – Clones correspond to equational theories: two algebras generate the same clone of term operations precisely when they satisfy the same identities. Consequently, clones provide an algebraic counterpart to varieties of algebras.
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Polymorphism clones – In constraint satisfaction problems and model theory, the clone of all polymorphisms of a relational structure (operations preserving its relations) is a central object of study.
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Lattice of clones – The collection of all clones on a fixed set $A$, ordered by inclusion, forms a complete algebraic lattice. This lattice is highly intricate; for finite $A$ it contains continuum many distinct clones.
Examples
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Full clone – The set of all finitary operations on $A$ is a clone; it is the greatest element of the clone lattice.
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Projection clone – The smallest clone consists solely of the projection maps; it is the least element of the lattice.
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Clone of a group – For a group $(G,\cdot)$, the term operations are precisely the group words interpreted in $G$; the corresponding clone contains operations such as the binary multiplication, unary inversion, and constants (the identity element), together with all compositions thereof.
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Clone of Boolean functions – On the two‑element set ${0,1}$, clones correspond to classes of Boolean functions closed under superposition; the classification of these clones (Post’s lattice) is a classical result.
Historical Note
The concept of a clone was introduced in the mid‑20th century in the context of universal algebra, notably by Emil Post (1941) for Boolean functions and later generalized by B. A. Burris and H. P. W. R. Rosenberg among others. It provides a unifying framework for studying the expressive power of algebraic operations across diverse mathematical structures.