Deciding absorption in relational structures
Libor Barto and Jakub Bul
Abstract. We prove that for ﬁnite, ﬁnitely related algebras, the concepts of an
absorbing subuniverse and a J´onsson absorbing subuniverse coincide. Consequently,
it is decidable whether a given subset is an absorbing subuniverse of the polymorphism
algebra of a given relational structure.
A ﬁnite algebra A is called ﬁnitely related if its clone of term operations
is equal to the polymorphism clone of some relational structure with ﬁnite
signature. In  it was proved that every ﬁnite, ﬁnitely related algebra with
J´onsson terms has a near unanimity term; consequently, it is decidable whether
a given relational structure has a near unanimity polymorphism. In the present
paper, we generalize this result and its consequence. A partial result in the
same direction already appeared in .
Absorption (J´onsson absorption, respectively) generalizes near unanimity
operations (J´onsson operations) in the following way. A subuniverse B or
a subalgebra B of an algebra A is absorbing, written B A, if A has an
idempotent term t (an absorption term) such that
t(B,B,...,B,A,B,...,B) ⊆ B
for every position of A. We also say that B absorbs A, that t witnesses the
absorption B A, etc.
A subuniverse B is called J´onsson absorbing, written B
A, if A has a se-
quence of (necessarily idempotent) terms d
, ..., d
(a J´onsson absorption
chain) such that
(B, A, B) ⊆ B for all i ≤ n,
(x, y, y) ≈ d
(x, x, y) for all i<n,
(x, y, z) ≈ x, and d
(x, y, z) ≈ z.
Presented by R. Willard.
Received January 19, 2016; accepted in ﬁnal form May 4, 2016.
2010 Mathematics Subject Classiﬁcation: Primary: 08A05; Secondary: 08B10, 08A70.
Key words and phrases: ﬁnitely related algebra, absorbing subalgebra, J´onsson absorbing
subalgebra, near unanimity, congruence distributivity.
Both authors were supported by the Grant Agency of the Czech Republic, grant 13-
01832S. The second author was also supported by the Polish National Science Centre (NCN)
Algebra Univers. 78 (2017) 3–18
Published online May 20, 2017
© Springer International Publishing 2017