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出 处:《数学进展》2008年第4期499-511,共13页Advances in Mathematics(China)
基 金:浙江省自然科学基金(No.Y605389);宁波市青年基金(No.2005A620032).
摘 要:首先讨论了De Morgan代数与剩余格的关系,并引入强De Morgan代数的概念,讨论了它的基本性质.随后,将著名的R_0蕴涵拓广到De Morgan代数上,称为广义R_0蕴涵;证明了添加广义R_0蕴涵和相应(?)算子后的De Morgan代数L成为剩余格的充要条件是L为强De Morgan代数,并由此引入DR_0代数的概念.接着,研究了DR_0代数与R_0代数的关系,证明了以下结论:Boole代数是DR_0代数;全序DR_0代数和全序R_0代数等价;DR_0代数是R_0代数当且仅当它满足预线性条件;无中点的DR_0代数是BL代数当且仅当它是Boole代数.最后,举例说明了非DR_0代数的R_0代数、以及非R_0代数的DR_0代数都是存在的.Firstly the relation between De Morgan algebras and regular residuated lattices are investigated, and the notion of strong De Morgan algebra is introduced, and its elementary properties are discussed. Secondly, the well-known Ro implication is developed to De Morgan algebras, which is called general R0 implication. And the necessary and sufficient condition is proved as following: a De Morgan algebra L with general R0 implication and corresponding operator becomes a residuated lattice if and only if L is a strong De Morgan algebra. By this result the notion of DRo algebras is introduced. Thirdly, the relations between DR0 algebras and R0 algebras are studied, and the following conclusions are proved: Boole algebras are DRo algebras; linear Ro algebra and linear DR0 algebra is equivalent; a DRo algebra is a Ro algebra if and only if it satisfies pre-lineax condition; DRo algebra without middle point is a BL-algebra if and only if it is a Boole algebra. Finally, some examples are given to show that there are Ro algebras which are not DRo algebras, and there are DR0 algebras which are not Ro algebras.
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