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NOTE ON LOGICS OF IDEMPOTENTS

NOTE ON LOGICS OF IDEMPOTENTS DEMONSTRATIO MATHEMATICAVol. XXXVIINo 22004Frantisek KatrnoskaNOTE ON LOGICS OF IDEMPOTENTSAbstract. The main result of this paper is the characterization of certain logicsof idempotents by Boolean semirings. Moreover some interesting examples are likewiseadded.1. IntroductionLet R be a ring with identity 1. Denote by U(R) the set of all idempotentsof the ring R. The following definition will play an important role in thesequel:DEFINITION 1. Let ( L , < , 0 , 1 , ' ) be a poset with 0 and 1 as the least andthe greatest element, respectively, and a unary operation ':L —• L (theorthocomplementation) such that:(i) p < q =• q' < p',(ii) &/)' = ?,( i i i ) p v j f = l , pe(iv) p < q'p,qtLpeLLpV q exists in L , p, q G L(v) p < q =• q = p V (p' A q), p,q G L.Then L will be called a logic or also an orthomodular poset. If L is also alattice, then L is called an orthomodular lattice.DEFINITION 2. Let L be a logic. A subset S of L is said to be a sublogic ofL if the following conditions are satisfied:(i) If peS then p' G S.(ii) If http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

NOTE ON LOGICS OF IDEMPOTENTS

Demonstratio Mathematica , Volume 37 (2): 8 – Apr 1, 2004

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References (5)

Publisher
de Gruyter
Copyright
© by František Katrnoška
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2004-0204
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XXXVIINo 22004Frantisek KatrnoskaNOTE ON LOGICS OF IDEMPOTENTSAbstract. The main result of this paper is the characterization of certain logicsof idempotents by Boolean semirings. Moreover some interesting examples are likewiseadded.1. IntroductionLet R be a ring with identity 1. Denote by U(R) the set of all idempotentsof the ring R. The following definition will play an important role in thesequel:DEFINITION 1. Let ( L , < , 0 , 1 , ' ) be a poset with 0 and 1 as the least andthe greatest element, respectively, and a unary operation ':L —• L (theorthocomplementation) such that:(i) p < q =• q' < p',(ii) &/)' = ?,( i i i ) p v j f = l , pe(iv) p < q'p,qtLpeLLpV q exists in L , p, q G L(v) p < q =• q = p V (p' A q), p,q G L.Then L will be called a logic or also an orthomodular poset. If L is also alattice, then L is called an orthomodular lattice.DEFINITION 2. Let L be a logic. A subset S of L is said to be a sublogic ofL if the following conditions are satisfied:(i) If peS then p' G S.(ii) If

Journal

Demonstratio Mathematicade Gruyter

Published: Apr 1, 2004

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