S-preclones and the Galois connection SPol–SInv, Part I

• Published in: Algebra universalis Vol. 85; no. 3
• Main Authors: Jipsen, Peter, Lehtonen, Erkko, Pöschel, Reinhard
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2024; Springer Nature B.V
• Subjects: Algebra; Mathematics; Mathematics and Statistics; Monoids; 06A15; Galois connection; 08A99; Preclone; Order-reversing map; 08A40; Partially ordered algebra; Order-preserving map; 06F99
• ISSN: 0002-5240; 1420-8911
• DOI: 10.1007/s00012-024-00863-7

We consider
S
-
operations
f
:
A
n
→
A
in which each argument is assigned a
signum
s
∈
S
representing a “property” such as being order-preserving or order-reversing with respect to a fixed partial order on
A
. The set
S
of such properties is assumed to have a monoid structure reflecting the behaviour of these properties under the composition of
S
-operations (e.g., order-reversing composed with order-reversing is order-preserving). The collection of all
S
-operations with prescribed properties for their signed arguments is not a clone (since it is not closed under arbitrary identification of arguments), but it is a preclone with special properties, which leads to the notion of
S
-
preclone
. We introduce
S
-
relations
ϱ
=
(
ϱ
s
)
s
∈
S
,
S
-
relational clones
, and a preservation property (
), and we consider the induced Galois connection
S
Pol
–
S
Inv
. The
S
-preclones and
S
-relational clones turn out to be exactly the closed sets of this Galois connection. We also establish some basic facts about the structure of the lattice of all
S
-preclones on
A
.