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Small clones and the projection property

# Small clones and the projection property,10.1007/s00012-010-0063-6,Algebra Universalis,Maurice Pouzet,Ivo G. Rosenberg

Small clones and the projection property
In 1986, the second author classified the minimal clones on a finite universe into five types. We extend this classification to infinite universes and to multiclones. We show that every non-trivial clone contains a “small” clone of one of the five types. From it we deduce, in part, an earlier result, namely that if $${\mathcal{C}}$$ is a clone on a universe A with at least two elements that contains all constant operations, then all binary idempotent operations are projections and some m-ary idempotent operation is not a projection for some m ≥ 3 if and only if there is a boolean group G on A for which $${\mathcal{C}}$$ is the set of all operations f(x 1, . . . , x n ) of the form $$a + {\sum_{i \in I}x_{i}}$$ for $${a \in A}$$ and $${I \subseteq \{1,\,.\,.\,.\,,n\}}$$.
Journal: Algebra Universalis - ALGEBRA UNIV , vol. 63, no. 1, pp. 37-44, 2010
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## Citation Context (2)

• ...´ lemma has fundamental consequences in universal algebra (see, e.g., [9, 13, 14, 16])...
• ...This classification was extended to clones of multioperations in [13], where ´ lemma was accordingly adjusted for multiprojections in the multi-valued case...

### Miguel Couceiro, et al. Generalizations of Swierczkowski's lemma and the arity gap of finite f...

• ...´ lemma has fundamental consequences in universal algebra (see, e.g., [9, 13, 14, 16])...
• ...This classification was extended to clones of multioperations in [13], where ´ lemma was accordingly adjusted for multiprojections in the multi-valued case...

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## Citations (2)

### Generalizations of Swierczkowski's lemma and the arity gap of finite functions(Citations: 6)

Journal: Discrete Mathematics - DM , vol. 309, no. 20, pp. 5905-5912, 2009

### Generalizations of Swierczkowski's lemma and the arity gap of finite functions(Citations: 1)

Published in 2007.