###### Options

# ω-categorical structures avoiding height 1 identities

Publikationstyp

Journal Article

Publikationsdatum

2021-01

Sprache

English

Volume

374

Issue

1

Start Page

327

End Page

350

Citation

Transactions of the American Mathematical Society 374 (1): 327-350 (2021-01)

Publisher DOI

Scopus ID

The algebraic dichotomy conjecture for Constraint Satisfaction Problems (CSPs) of reducts of (infinite) finitely bounded homogeneous structures states that such CSPs are polynomial-time tractable if the model-complete core of the template has a pseudo-Siggers polymorphism, and is NP-complete otherwise. One of the important questions related to the dichotomy conjecture is whether, similarly to the case of finite structures, the condition of having a pseudo-Siggers polymorphism can be replaced by the condition of having polymorphisms satisfying a fixed set of identities of height 1, i.e., identities which do not contain any nesting of functional symbols. We provide a negative answer to this question by constructing for each nontrivial set of height 1 identities a structure within the range of the conjecture whose polymorphisms do not satisfy these identities, but whose CSP is tractable nevertheless. An equivalent formulation of the dichotomy conjecture characterizes tractability of the CSP via the local satisfaction of nontrivial height 1 identities by polymorphisms of the structure. We show that local satisfaction and global satisfaction of nontrivial height 1 identities differ for ω-categorical structures with less than doubly exponential orbit growth, thereby resolving one of the main open problems in the algebraic theory of such structures.