Abstract
We show that every generator, in a certain set of generators for the variety of reflexive near unanimity graphs, admits a semilattice polymorphism. We then find a retract of a product of such graphs (paths, in fact) that has no semilattice polymorphism. This verifies for reflexive graphs that the variety of graphs with semilattice polymorpisms does not contain the variety of graphs with near-unanimity, or even $3$-ary near-unanimity polymorphisms.
Highlights
For relational structures such as graphs, the existence of relation preserving operations, or polymorphisms, satisfying various identities has been of great interest recently due to its relation to the complexity of the problem of deciding whether or not there is a homomorphism between given structures
In this paper we look at near-unanimity (NU), and semilattice (SL) polymorphisms on reflexive graphs
We talk of totally symmetric idempotent (TSI) polymorphisms
Summary
For relational structures such as graphs, the existence of relation preserving operations, or polymorphisms, satisfying various identities has been of great interest recently due to its relation to the complexity of the problem of deciding whether or not there is a homomorphism between given structures. We verified that SL is not closed under retraction, so though it is closed under products, it is not a variety This shows that it is different from the classes of graphs admitting TSI or NU polymorphisms. Proposition 7 is not a ploy for building tension before surprising the reader with Theorem 9 It was observed in [4] that every NU structure H is the retraction of some universal structure UTSI(H) which can be shown to admit an SL polymorphism.
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