Incidence algebras
Content created by Garrett Figueroa and Egbert Rijke.
Created on 2024-11-14.
Last modified on 2025-01-25.
module order-theory.incidence-algebras where
Imports
open import commutative-algebra.commutative-rings open import foundation.dependent-pair-types open import foundation.inhabited-types open import foundation.universe-levels open import foundation-core.cartesian-product-types open import order-theory.interval-subposets open import order-theory.locally-finite-posets open import order-theory.posets
Idea
For a locally finite poset P
and
commutative ring R
, there is a
canonical R
-associative algebra whose underlying R
-module are the set-maps
from the nonempty intervals of P
to R
(which we constructify as the inhabited intervals), and whose multiplication is
given by a “convolution” of maps. This is the incidence algebra of P
over
R
.
Definition
module _ {l1 l2 l3 : Level} (P : Poset l1 l2) (loc-fin : is-locally-finite-Poset P) (x y : type-Poset P) (R : Commutative-Ring l3) where interval-map : UU (l1 ⊔ l2 ⊔ l3) interval-map = inhabited-interval P → type-Commutative-Ring R
WIP: complete this definition after R-modules have been defined. Defining convolution of maps would be aided as well with a lemma on ‘unordered’ addition in abelian groups over finite sets.
Recent changes
- 2025-01-25. Garrett Figueroa. No-postfix-projections (#1240).
- 2024-11-14. Garrett Figueroa and Egbert Rijke. Incidence algebras (#1221).