The objective of this paper is to introduce a method for constructing a minimal lattice-valued tree automaton with membership values in a totally ordered lattice (in short \(\mathcal {LTA}),\) based on the solvability of a system of fuzzy polynomial equations. Since the minimization problem strongly depends on the equivalence problem, at first, the equivalence problem is examined. For this purpose, the notion of h-equivalence is defined, and a necessary and sufficient condition for the equivalence between two \(\mathcal {LTA}s\) is provided. It is shown that the equivalence problem of \(\mathcal {LTA}s\) is decidable. In the minimization problem, the following question is replied: given an \(\mathcal {LTA}\) \(\mathbb {A}\) and a positive integer k, is there an \(\mathcal {LTA}\) with k states equivalent to \(\mathbb {A}?\) Decidability of the minimization problem is demonstrated, and an approach to return a minimal \(\mathcal {LTA}\) equivalent to the original one is presented. Also, the time complexity of the proposed algorithm is analyzed. Finally, some examples are presented to clarify the minimization problem.

本文的目的是基于模糊系统的可解性，介绍一种在完全有序晶格（简称\（\ mathcal {LTA}）\）中构造具有隶属值的最小晶格值树自动机的方法。多项式方程。由于最小化问题主要取决于等价问题，因此首先要研究等价问题。为此，定义了h等价的概念，并为两个\（\ mathcal {LTA} s \）之间的等价提供了充要条件。结果表明，\（\ mathcal {LTA} s \）的等价问题是可判定的。在最小化问题中，将回答以下问题：给定一个\（\ mathcal {LTA} \） \（\ mathbb {A} \）和一个正整数k，是否存在一个\（\ mathcal {LTA} \）且k个状态等于\（\ mathbb {A}？\\），所以证明了最小化问题的可判定性，并提出了一种返回等于原始\（\ mathcal {LTA} \）的最小\（\ mathcal {LTA} \）的方法。此外，分析了所提算法的时间复杂度。最后，给出一些例子来阐明最小化问题。