Abstract Inspired by the open problems “How to define the notions of fantastic filters and states in EQ-algebras” in [LIU, L. Z.—ZHANG, X. Y.: Implicative and positive implicative prefilters of EQ-algebras, J. Intell. Fuzzy Syst. 26 (2014), 2087–2097], we introduce the notions of fantastic filters and investigate the existence of Bosbach states and Riečan states on EQ-algebras by use of fantastic filters. Firstly, we prove that a residuated EQ-algebra has a Bosbach state if and only if it has a fantastic filter. We also establish that a good EQ-algebra has a state-morphism if and only if it has a prime fantastic filter. Furthermore, we introduce the notion of QI-EQ-algebras and obtain the necessary and sufficient condition for a residuated QI-EQ-algebra having Riečan states. Finally, we introduce the notion of semi-divisible EQ-algebras and give an example of a semi-divisible residuated EQ-algebra, which is not a semi-divisible residuated lattice. We also prove that every semi-divisible residuated EQ-algebra admits Riečan states. These works generalize a series of existing results about existence of states in several algebras, such as residuated lattices, NM-algebras, MTL-algebras, BL-algebras and so on.

中文翻译：

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EQ-代数上状态的存在性

摘要 受 [LIU, LZ-ZHANG, XY: Implicative and positive Implicative prefilters of EQ-algebras, J. Intell. 模糊系统。26 (2014), 2087–2097]，我们介绍了梦幻般的滤波器的概念，并通过使用梦幻般的滤波器研究了 EQ 代数上 Bosbach 状态和 Riečan 状态的存在。首先，我们证明残差 EQ 代数具有 Bosbach 状态当且仅当它具有出色的滤波器。我们还确定一个好的 EQ 代数具有状态态射当且仅当它有一个主要的梦幻般的过滤器。此外，我们引入了 QI-EQ-代数的概念，并获得了具有 Riečan 状态的剩余 QI-EQ-代数的充要条件。最后，我们介绍了半可分 EQ-代数的概念，并给出了一个半可分剩余 EQ-代数的例子，它不是一个半可分的剩余格。我们还证明了每个半可分的残差 EQ 代数都承认 Riečan 状态。这些工作概括了一些关于剩余格、NM-代数、MTL-代数、BL-代数等代数中状态存在的现有结果。