POWERSET OPERATOR FOUNDATIONS FOR CATALG FUZZY SET THEORIES

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SOLOVYOV S.A.

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Journal: Iranian Journal of Fuzzy Systems Year:2011 | Volume:8 | Issue:2 Page(s): 1-46

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Information Journal Paper

Title

POWERSET OPERATOR FOUNDATIONS FOR CATALG FUZZY SET THEORIES

Author(s)

SOLOVYOV S.A. | Issue Writer Certificate

Keywords

(BACKWARD; FORWARD) POWERSET (OPERATOR; THEORY)Q2

(BI;CO)PRODUCT (COMPOSITE) (VARIETY-BASED) TOPOLOGICAL (SPACE; SYSTEM)Q2

(CONVENIENT) VARIETYQ2

GROUND CATEGORYQ2

(L-) FUZZY SETQ2

(LOCALIC) ALGEBRAQ2

MONADIC (CATEGORY; LOGIC)Q2

ORDERED CATEGORYQ2

POINTED CATEGORYQ2

QUANTALOIDQ2

(SEMI-) QUANTALEQ2

(UNITAL) (QUANTALE; RING) (ALGEBRA; MODULE)Q2

ZERO (MORPHISM; OBJECT)Q2

Abstract

The paper sets forth in detail categorically-algebraic or catalg foundations for the operations of taking the image and preimage of (fuzzy) sets called forward and backward powerset operators. Motivated by an open question of S. E.Rodabaugh, we construct a monad on the category of sets, the algebras of which generate the xed-basis forward powerset operator of L. A. Zadeh. On the next step, we provide a direct lift of the backward powerset operator using the notion of categorical biproduct. The obtained framework is readily extended to the variable-basis case, justifying the powerset theories currently popular in the fuzzy community. At the end of the paper, our general variety-based setting postulates the requirements, under which a (CONVENIENT) VARIETY-based powerset theory can be developed, suitable for employment in all areas of fuzzy mathematics dealing with fuzzy powersets, including fuzzy algebra, logic and topology.

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APA: Copy

SOLOVYOV, S.A.. (2011). POWERSET OPERATOR FOUNDATIONS FOR CATALG FUZZY SET THEORIES. IRANIAN JOURNAL OF FUZZY SYSTEMS, 8(2), 1-46. SID. https://sid.ir/paper/113384/en

Vancouver: Copy

SOLOVYOV S.A.. POWERSET OPERATOR FOUNDATIONS FOR CATALG FUZZY SET THEORIES. IRANIAN JOURNAL OF FUZZY SYSTEMS[Internet]. 2011;8(2):1-46. Available from: https://sid.ir/paper/113384/en

IEEE: Copy

S.A. SOLOVYOV, “POWERSET OPERATOR FOUNDATIONS FOR CATALG FUZZY SET THEORIES,” IRANIAN JOURNAL OF FUZZY SYSTEMS, vol. 8, no. 2, pp. 1–46, 2011, [Online]. Available: https://sid.ir/paper/113384/en

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