We investigate when fiber products of lattices are finitely generated and obtain a new characterization of bounded lattice homomorphisms onto lattices satisfying a property we call Dean’s condition (D) which arises from Dean’s solution to the word problem for finitely presented lattices. In particular, all finitely presented lattices and those satisfying Whitman’s condition satisfy (D). For lattice epimorphisms [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text] are finitely generated and [Formula: see text] satisfies (D), we show the following: If [Formula: see text] and [Formula: see text] are bounded, then their fiber product (pullback) [Formula: see text] is finitely generated. While the converse is not true in general, it does hold when [Formula: see text] and [Formula: see text] are free. As a consequence, we obtain an (exponential time) algorithm to decide boundedness for finitely presented lattices and their finitely generated sublattices satisfying (D). This generalizes an unpublished result of Freese and Nation.

中文翻译：

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晶格的有界同态和有限生成的纤维积

我们研究了何时有限地生成晶格的纤维产品，并获得了晶格上的有界晶格同态的新表征，该晶格满足我们称为迪恩条件 (D) 的属性，该属性源自迪恩对有限呈现晶格的单词问题的解决方案。特别是，所有有限呈现的格和满足惠特曼条件的格都满足 (D)。对于格外同态 [Formula: see text], [Formula: see text], 其中 [Formula: see text], [Formula: see text] 是有限生成的并且 [Formula: see text] 满足 (D)，我们显示以下：如果[公式：见文]和[公式：见文]是有界的，那么它们的纤维乘积（回拉）[公式：见文]是有限生成的。虽然反之一般不成立，但当 [公式：见文本] 和 [公式：见文本] 是自由的时，它确实成立。作为结果，我们获得了一个（指数时间）算法来确定有限呈现的格及其有限生成的满足（D）的子格的有界性。这概括了 Freese 和 Nation 的未发表结果。