单生成拟簇与剩余结构
Singly generated quasivarieties and residuated structures
摘要：
代数的拟簇K具有联合嵌入性质（JEP）当且仅当它由单个代数A生成。它是结构完备的当且仅当K中的自由可数生成代数可以作为A。这种要求的一个结果，称为“被动结构完备性”（PSC），是K的非平凡成员都满足相同的存在肯定句。证明了若K是PSC，则它仍有JEP;若K有JEP且其非平凡成员缺少平凡子代数，则其相对简单的成员都属于其中一个生成的泛类.在这些条件下，如果K是相对半单的，则它是由一个K-单代数生成的。它是极小拟簇，如果它是PSC但不能统一某些有限集方程。我们还证明了具有有限个非平凡成员的有限型拟簇是PSC当且仅当它的非平凡成员有一个公共收缩核。然后将该理论应用于各种德摩根幺半群，在那里我们隔离的子（准）品种是PSC和那些有JEP，而扔在那些结构完整的新鲜光。这些结果对直觉逻辑和关联逻辑的可拓格具有启发意义。
英文摘要：
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.
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