Operads of Poset Matrices

Abstract

This paper establishes operad structures on the collection of poset matrices by introducing a new framework of partial composition operations. Extending the combinatorial setting of naturally labelled posets, we define several partial compositions that serve as basic tools for constructing poset matrices of arbitrary size. We prove that three of these operations satisfy the axioms of operads, thereby giving the operad structure to the set of poset matrices. We further characterize the dual operations and provide explicit combinatorial interpretations of these constructions in terms of naturally labeled posets. In addition, we address some open questions on poset enumeration via the operads of poset matrices.