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arXiv · 2502.06912

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

A. N. Bhavale
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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 $n \geq 4$ elements, having nullity $k \geq 1$, and containing $2 \leq r \leq 2k$ reducible elements. Secondly we enumerate all non-isomorphic RC-lattices on $n \geq 4$ elements, having nullity $k \geq 1$. This work is in respect of Birkhoff's open problem of enumerating all finite lattices on $n$ elements.

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A. N. Bhavale. 2025-02-10. Enumeration of lattices of nullity $k$ and containing $r$ comparable reducible elements. https://arxiv.org/abs/2502.06912

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