What question did this study set out to answer?

To develop the algebra of filters on finite posets and explore their connections to other mathematical structures.

March 30, 2026Open Access

Algebras of filters on finite posets

Key Points

To develop the algebra of filters on finite posets and explore their connections to other mathematical structures.
Constructed the algebra of filters on finite posets over a commutative ring.
Defined multiplication via ideal-filter decompositions.
Established an injective homomorphism into the Rota incidence algebra.
Studied the group of units and its structure as a product of unit groups and special linear groups.
Introduced SU algebras and computed their homological invariants and Hochschild (co)homology.
Demonstrated that Butcher's algebra is a special case of the developed algebra of filters.
Showed that the group of units has a strict nilpotent structure and unique factorizations.
Computed the Poincaré series and identified invariant ideals, establishing relationships between algebras.
Constructed essential Hopf algebra structures linking with combinatorial aspects.

Abstract

We develop a systematic theory of the algebra of filters (, ) on a finite poset (, ) over a commutative ring~. and embedding. The multiplication is defined via ideal-filter decompositions. We show that Butcher's algebra on finite labeled forests is a special case. Thus (, ) extends the forest construction to arbitrary finite posets and retains its links with numerical integration and the Connes--Kreimer Hopf algebra. We construct an injective -algebra homomorphism (, ) (2^, ) into the Rota incidence algebra of the Boolean lattice, and show that when is an antichain, (, ) is isomorphic to the ring of arithmetical functions under unitary convolution. of units. We study the group of units (, ) = (, ) ^. We prove (, ) () (, ), where (, ) consists of elements with augmentation~1. Every element of (, ) admits a unique factorization as a product of elementary invertible functions, (, ) is nilpotent of class at most~||, and we give explicit finite presentations over and finite cyclic rings. invariants. We show the defining relations of () form a quadratic Gr\"obner basis, so _ (, ) is Koszul for any field~. We introduce the class of SU~algebras, prove it is closed under quadratic duality, and show _ () is a strictly stronger poset invariant than the Stanley--Reisner ring. We construct a minimal free biresolution, compute the Poincar\'e series, and determine the invariant ideals. Using the biresolution we compute the Hochschild (co) homology, identify the Ext algebra with the Koszul dual, and characterize the center Z ( () ) = ⁰ ( (), () ) combinatorially. algebra structures. We construct a graded connected commutative Hopf algebra (, ) whose coproduct is defined by ideal decompositions. Its character group is naturally isomorphic to (, ) ^op, linking the Hopf-algebraic and unit-group viewpoints. We define a connected quotient ᶜ (, ) and show that for rooted forests it recovers the Connes--Kreimer coproduct. Finally, we prove functoriality with respect to order embeddings and determine the primitive and cocommutative cases.

Read Full Paperexternally

Mark Helpful

Bookmark

Relay

View Full Paper