关于簇和拟簇的强结构完备性
On Strong Structural Completeness of Varieties and Quasivarieties
AI总结：
研究代数框架下的强结构完备性，证明有限型且具有CEP的拟簇若由有限代数生成且包含无限不可约代数则非强结构完备，并给出强原始性的判定准则。
摘要：
我们在代数背景下研究无穷意义下的结构完备性（强结构完备性）。一个簇是结构完备的（SCpl），如果它作为拟簇由其自由代数生成；它是强结构完备的（SSCpl），如果它作为预簇由其自由代数生成。一个拟簇是SSCpl，如果它作为预簇由其自由代数生成。我们证明，每个有限型且具有CEP的拟簇，如果由有限代数生成且包含一个无限不可约代数，则它不是SSCpl。此外，一个由有限代数生成的有限型同余交半分配簇是SSCpl当且仅当它是表格的。因此，Dummett逻辑和Medvedev逻辑是SCpl但不是SSCpl。一个簇是原始的，如果它是SCpl且其所有子簇都是SCpl；它是强原始的，如果它是SSCpl且其所有子簇都是SSCpl。我们证明，在有限型的原始同余分配簇中，表格子簇（且只有这些子簇）是强原始的。这一观察也给出了强原始性的一个判定准则。
英文摘要：
We study structural completeness in the infinitary sense (strong structural completeness) in an algebraic setting. A variety is structurally complete (SCpl) if it is generated, as a quasivariety, by its free algebras, and it is strongly structurally complete (SSCpl) if it is generated, as a prevariety, by its free algebras. A quasivariety is SSCpl if it is generated, as a prevariety, by its free algebras. We prove that every quasivariety of finite type with the CEP that is generated by finite algebras and contains an infinite irreducible algebra is not SSCpl. Moreover, every congruence meet-semidistributive variety of finite type generated by finite algebras is SSCpl if and only if it is tabular. Thus, Dummett's and Medvedev's logics are SCpl but not SSCpl. A variety is primitive if it is SCpl and all its subvarieties are SCpl; it is strongly primitive if it is SSCpl and all its subvarieties are SSCpl. We prove that in primitive congruence-distributive varieties of finite type, the tabular subvarieties, and only those, are strongly primitive. This observation also yields a criterion for strong primitivity.
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