A quasivariety K of algebras has the joint embedding property (JEP) iff it is generated by a single algebra A. It is structurally complete iff the free countably generated algebra in K can serve as A. A consequence of this demand, called "passive structural completeness" (PSC), is that the nontrivial members of K all satisfy the same existential positive sentences. We prove that if K is PSC then it still has the JEP, and if it has the JEP and its nontrivial members lack trivial subalgebras, then its relatively simple members all belong to the universal class generated by one of them. Under these conditions, if K is relatively semisimple then it is generated by one K-simple algebra. It is a minimal quasivariety if, moreover, it is PSC but fails to unify some finite set of equations. We also prove that a quasivariety of finite type, with a finite nontrivial member, is PSC iff its nontrivial members have a common retract. The theory is then applied to the variety of De Morgan monoids, where we isolate the sub(quasi)varieties that are PSC and those that have the JEP, while throwing fresh light on those that are structurally complete. The results illuminate the extension lattices of intuitionistic and relevance logics.

中文翻译：

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单独生成的拟变体和剩余结构

代数的拟变体 K 具有联合嵌入属性 (JEP)，当它由单个代数 A 生成。当 K 中的自由可数生成代数可以作为 A 时，它在结构上是完备的。这种需求的结果，称为“被动结构完整性”（PSC），是 K 的非平凡成员都满足相同的存在肯定句。我们证明如果 K 是 PSC 那么它仍然有 JEP，如果它有 JEP 并且它的非平凡成员缺少平凡子代数，那么它的相对简单的成员都属于由其中之一生成的通用类。在这些条件下，如果 K 是相对半简单的，那么它是由一个 K-简单代数生成的。此外，如果它是 PSC 但未能统一某些有限方程组，则它是最小拟方差。我们还证明了一个有限类型的拟变体，具有有限的非平凡成员，是 PSC 当当其非平凡成员有一个共同的缩回。然后将该理论应用于各种 De Morgan 幺半群，在那里我们分离出 PSC 和具有 JEP 的亚（准）变种，同时对结构完整的亚种进行新的阐述。结果阐明了直觉逻辑和相关逻辑的扩展格。