半群模簇的平面性秩
Planarity ranks of modular varieties of semigroups
AI总结：
本文计算了所有半群模簇的平面性秩的精确值，证明其不超过3，并利用机器证明与图论准则完成验证。
摘要：
半群簇的平面性秩是指该簇的自由半群中，使得该半群具有平面凯莱图的最大生成元数目。自L.M.马尔蒂诺夫提出描述半群簇平面性秩的问题以来，该方向已取得许多具体结果。半群的模簇是指其子簇格为模格的半群簇。本文计算了所有可能的半群模簇的无限可数集合中平面性秩的精确值。结果表明这些值不超过3。证明中主要使用机器计算。使用Prover9和Mace4来检验由大量恒等式定义的簇的自由半群中元素的相等性。为证明图的非平面性，使用庞特里亚金-库拉托夫斯基准则，并间接使用科林·德·韦尔迪埃不变量来证明平面性。
英文摘要：
By the planarity rank of a semigroup variety we mean the largest number of generators of a free semigroup of a variety with respect to which the semigroup admits a planar Cayley graph. Since the time when L.M.Martynov formulated the problem of describing the planarity ranks of semigroup varieties, many specific results have been obtained in this direction. A modular variety of semigroups is a variety of semigroups with a modular lattice of subvarieties. In this paper, we calculate the exact values of the planarity ranks of an infinite countable set of all possible modular varieties of semigroups. It turns out that these values do not exceed 3. Machine calculations are mostly used in the proof. Prover9 and Mace4 are used to check the equalities of elements of free semigroups of varieties defined by a large number of identities. To prove the non-planarity of graphs, the Pontryagin-Kuratovsky criterion is used, and the Colin de Verdiere invariant is indirectly used to justify planarity.
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