Abstract
In the first part of this paper we present a new family of finite bounded posets whose clones of monotone operations are not finitely generated. The proofs of these results are analogues of those in the famous paper of Tardos. Another interesting family of finite posets from the finite generability point of view is the family of locked crowns. To decide whether the clone of a locked crown where the crown is of at least six elements is finitely generated or not one needs to go beyond the scope of Tardos’s proof. Although our investigations are not conclusive in this direction, they led to the results in the second part of the paper. We call a monotone operation ascending if it is greater than or equal to some projection. We prove that the clones of bounded posets are generated by certain ascending idempotent monotone operations and the 0 and 1 constant operations. A consequence of this result is that if the clone of ascending idempotent operations of a finite bounded poset is finitely generated, then its clone is finitely generated as well. We provide an example of a half bounded finite poset whose clone of ascending idempotent operations is finitely generated but whose clone is not finitely generated. Another interesting consequence of our result is that if the clone of a finite bounded poset is finitely generated, then it has a three element generating set that consists of an ascending idempotent monotone operation and the 0 and 1 constant operations.
Highlights
Let F be a set of operations on a set A
We prove that the clones of bounded posets are generated by certain ascending idempotent monotone operations and the 0 and 1 constant operations
A consequence of this result is that if the clone of idempotent operations of a finite bounded poset is finitely generated, its clone is finitely generated as well. Another interesting consequence of our result is that if the clone of a finite bounded poset is finitely generated, it has a three element generating set that consists of an ascending idempotent monotone operation and the 0 and 1 constant operations
Summary
Let F be a set of operations on a set A. In [3] Demetrovics and Ronyai proved that the clone of any crown is finitely generated It is well known, on the other hand, that the idempotent clone of any crown contains only projections, its clone does not contain a near unanimity operation. On the other hand, that the idempotent clone of any crown contains only projections, its clone does not contain a near unanimity operation In his famous paper [5] Tardos proved that the clone of the eight element poset T in Fig. 1 is not finitely generated. Our result does not extend to half bounded finite posets: we prove that the clone of ascending idempotent operations of H is finitely generated but, as we mentioned above, the clone of H is not finitely generated. It remains an open question whether the clone of Dk, k ≥ 3, is finitely generated
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