Projectivity in (bounded) commutative integral residuated lattices

IF 0.6 4区 数学 Q3 MATHEMATICS Algebra Universalis Pub Date : 2022-11-29 DOI:10.1007/s00012-022-00798-x
Paolo Aglianò, Sara Ugolini
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引用次数: 5

Abstract

In this paper, we study projective algebras in varieties of (bounded) commutative integral residuated lattices. We make use of a well-established construction in residuated lattices, the ordinal sum, and the order property of divisibility. Via the connection between projective and splitting algebras, we show that the only finite projective algebra in \(\mathsf {{FL}_{ew}}\) is the two-element Boolean algebra. Moreover, we show that several interesting varieties have the property that every finitely presented algebra is projective, such as locally finite varieties of hoops. Furthermore, we show characterization results for finite projective Heyting algebras, and finitely generated projective algebras in locally finite varieties of bounded hoops and BL-algebras. Finally, we connect our results with the algebraic theory of unification.

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(有界)交换积分剩余格的射影性
本文研究了各种(有界)交换积分剩余格中的射影代数。我们在剩余格中使用了一个已建立的构造,序数和,以及可分性的序性质。通过射影代数和分裂代数之间的联系,我们证明了\(\mathsf)中唯一的有限射影代数{{FL}_{ew}}\)是二元布尔代数。此外,我们还证明了几个有趣的变种具有每个有限表示代数都是射影的性质,例如环的局部有限变种。此外,我们给出了有限投影Heyting代数的刻画结果,以及局部有限有界环和BL代数中的有限生成投影代数。最后,我们将我们的结果与统一的代数理论联系起来。
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来源期刊
Algebra Universalis
Algebra Universalis 数学-数学
CiteScore
1.00
自引率
16.70%
发文量
34
审稿时长
3 months
期刊介绍: Algebra Universalis publishes papers in universal algebra, lattice theory, and related fields. In a pragmatic way, one could define the areas of interest of the journal as the union of the areas of interest of the members of the Editorial Board. In addition to research papers, we are also interested in publishing high quality survey articles.
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