Finite sublattices in the lattice of clones

1994 ◽  
Vol 33 (5) ◽  
pp. 287-306 ◽  
Author(s):  
A. A. Bulatov
Keyword(s):  
Lattice Of Clones

scholarly journals Ideal clones: Solution to a problem of Czédli and Heindorf

2010 ◽  
Vol 47 (4) ◽  
pp. 419-429
Author(s):  
Martin Goldstern ◽  
Michael Pinsker
Keyword(s):  
Affirmative Answer ◽  
Countably Infinite ◽  
Infinite Set ◽  
Lattice Of Clones

Given an infinite set X and an ideal I of subsets of X, the set of all finitary operations on X which map all (powers of) I-small sets to I-small sets is a clone. In [2], G. Czédli and L. Heindorf asked whether or not for two particular ideals I and J on a countably infinite set X, the corresponding ideal clones were a covering in the lattice of clones. We give an affirmative answer to this question.

ON DUALIZING CLONES AS LAWVERE THEORIES

2006 ◽  
Vol 16 (04) ◽  
pp. 657-687 ◽  
Author(s):  
DRAGAN MAŠULOVIĆ
Keyword(s):  
Boolean Algebra ◽  
Lattice Of Clones

In this paper we treat clones as Lawvere theories and then dualize them as categories, rather than as single objects of a category of algebras. The approach applies only to some primitive-positive clones, but in return, a structure with somewhat surprising properties is obtained. To illustrate the method, we thoroughly investigate the lattice of clones of operations over a finite boolean algebra.

scholarly journals The lattice of monomial clones on finite fields

2021 ◽  
Vol 82 (4) ◽  
Author(s):  
Sebastian Kreinecker
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Affine Algebras ◽  
Lattice Of Clones

AbstractWe investigate the lattice of clones that are generated by a set of functions that are induced on a finite field $${\mathbb {F}}$$ F by monomials. We study the atoms and coatoms of this lattice and investigate whether this lattice contains infinite ascending chains, or infinite descending chains, or infinite antichains.We give a connection between the lattice of these clones and semi-affine algebras. Furthermore, we show that the sublattice of idempotent clones of this lattice is finite and every idempotent monomial clone is principal.

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