The varieties generated by 3-hypergraph semirings
Yuanfan Zhuo, Xingliang Liang, Yanan Wu, Xianzhong Zhao
TL;DR
The paper studies varieties generated by 3-hypergraph semirings in ai-semiring settings. It constructs 3-hypergraph semigroups SG_H and the associated semirings S_H, proving S_H are flat, 0-cancellative, nonfinitely based, and subdirectly irreducible, with a reduction that every V(S_H) matches V(S_H') for some 3-uniform H'. It then shows that varieties coming from 2-robustly strong 3-colorable 3-colorable 3-uniform hypergraphs coincide with V(S_c(abc)), and examines beam-type, fan-type, and nested-type hypergraphs to produce infinite chains of subvarieties, illustrating a rich lattice structure. The results collectively reduce the study to 3-uniform cases, identify key generating varieties such as V(S_c(abc)) and V(S_c(abcd)), and demonstrate extensive diversity in the subvariety lattice of 3-uniform hypergraph semirings.
Abstract
In this paper the 3-hypergraph semigroups and 3-hypergraph semirings from 3-hypergraphs $\mathbb{H}$ are introduced and the varieties generated by them are studied. It is shown that all 3-hypergraph semirings $S_{\scriptscriptstyle \mathbb{H}}$ are nonfinitely based and subdirectly irreducible. Also, it is proved that each variety generated by 3-hypergraph semirings is equal to a variety generated by 3-uniform hypergraph semirings. It is well known that both variety $\mathbf{V}(S_c(abc))$ (see, J. Algebra 611: 211--245, 2022 and J. Algebra 623: 64--85, 2023) and variety $\mathbf{V}(S_{\scriptscriptstyle \mathbb{H}})$ play key role in the theory of variety of ai-semirings, where 3-uniform hypergraph $\mathbb{H}$ is a 3-cycle. They are shown that each variety generated by 2-robustly strong 3-colorable 3-uniform hypergraph semirings is equal to variety $\mathbf{V}(S_c(abc))$, and each variety generated by so-called beam-type hypergraph semirings or fan-type hypergraph semirings is equal to the variety $\mathbf{V}(S_{\scriptscriptstyle \mathbb{H}})$ generated by a 3-uniform 3-cycle hypergraph semiring $S_{\scriptscriptstyle \mathbb{H}}$. Finally, an infinite ascending chain is provided in the lattice of subvarieties of the variety generated by all 3-uniform hypergraph semirings. This implies that the variety generated by all 3-uniform hypergraph semirings has infinitely many subvarieties.