Enumeration of lattices of nullity $k$ and containing $r$ comparable reducible elements

Abstract

In 2002 Thakare et al.\ counted non-isomorphic lattices on $n$ elements, having nullity up to two. In 2020 Bhavale and Waphare introduced the concept of RC-lattices as the class of all lattices in which all the reducible elements are comparable. In this paper, we enumerate all non-isomorphic RC-lattices on $n$ elements. For this purpose, firstly we enumerate all non-isomorphic RC-lattices on $n\ge 4$ elements, having nullity $k\ge 1$, and containing $2\le r\le 2k$ reducible elements. Secondly we enumerate all non-isomorphic RC-lattices on $n\ge 4$ elements, having nullity $k\ge 1$. This work is in respect of Birkhoff's open problem of enumerating all finite lattices on $n$ elements.

11 pages. arXiv admin note: substantial text overlap with arXiv:2407.06736