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Published in:
2014 | OriginalPaper | Chapter
5. Classes of Non-commutative Residuated Structures
Author : Lavinia Corina Ciungu
Published in: Non-commutative Multiple-Valued Logic Algebras
Publisher: Springer International Publishing
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Abstract
In this chapter we study special classes of non-commutative residuated structures: local, perfect and Archimedean structures. The local bounded pseudo-BCK(pP) algebras are characterized in terms of primary deductive systems, while the perfect pseudo-BCK(pP) algebras are characterized in terms of perfect deductive systems. One of the main results consists of proving that the radical of a bounded pseudo-BCK(pP) algebra is normal. We also prove that any linearly ordered pseudo-BCK(pP) algebra and any locally finite pseudo-BCK(pP) algebra are local. Other results state that any local FL
w
-algebra and any locally finite FL
w
-algebra are directly indecomposable. The classes of Archimedean and hyperarchimedean FL
w
-algebras are introduced and it is proved that any locally finite FL
w
-algebra is hyperarchimedean and any hyperarchimedean FL
w
-algebra is Archimedean.