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HYPERIDENTITIES AND HYPERSUBSTITUTIONS IN THE VARIETY OF SYMMETRIC, IDEMPOTENT, ENTROPIC GROUPOIDS

HYPERIDENTITIES AND HYPERSUBSTITUTIONS IN THE VARIETY OF SYMMETRIC, IDEMPOTENT, ENTROPIC GROUPOIDS DEMONSTRATIO MATHEMATICAVol. XXXIINo 41999Sr. Arworn, Κ. Denecke*HYPERIDENTITIES A N D HYPERSUBSTITUTIONSIN T H E VARIETY OF SYMMETRIC, IDEMPOTENT,ENTROPIC GROUPOIDSAbstract. In this paper we determine the structure of the groupoid of normal form hypersubstitutions with respect to the variety of symmetric, idempotent, entropic groupoids,describe the monoid of all proper hypersubstitutions, and ask which identities are satisfiedas hyperidentities.1. IntroductionSymmetric, idempotent, and entropic groupoids are algebras of type 2satisfying the medial, the idempotent, and the so-called symmetric identity(z • y) • y ~ x- Following the paper [Ros; 87] we will denote them as SIEgroupoids, i.e.SIE = Mod{(χ • y) • y « χ, χ • χ & χ, (χ • y) • (u · ν) « (χ · u) • (y · υ)}.The variety SIE is a subvariety of the variety of binary modes which is defined only by the medial and the idempotent law. For instance, the reflectionof a point χ at a point y of the real line, i.e. χ * y := 2y — χ satisfies ailaxioms of SIE since(x * y) * y = 2y - (2y - x) = x,χ ^ χ — 2«π χ — χ j(x *y) * (u*v) = 2(2v — u) — http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

HYPERIDENTITIES AND HYPERSUBSTITUTIONS IN THE VARIETY OF SYMMETRIC, IDEMPOTENT, ENTROPIC GROUPOIDS

Demonstratio Mathematica , Volume 32 (4): 10 – Jan 1, 1999

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References (3)

Publisher
de Gruyter
Copyright
© by Sr. Arworn
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1999-0402
Publisher site
See Article on Publisher Site

Abstract

DEMONSTRATIO MATHEMATICAVol. XXXIINo 41999Sr. Arworn, Κ. Denecke*HYPERIDENTITIES A N D HYPERSUBSTITUTIONSIN T H E VARIETY OF SYMMETRIC, IDEMPOTENT,ENTROPIC GROUPOIDSAbstract. In this paper we determine the structure of the groupoid of normal form hypersubstitutions with respect to the variety of symmetric, idempotent, entropic groupoids,describe the monoid of all proper hypersubstitutions, and ask which identities are satisfiedas hyperidentities.1. IntroductionSymmetric, idempotent, and entropic groupoids are algebras of type 2satisfying the medial, the idempotent, and the so-called symmetric identity(z • y) • y ~ x- Following the paper [Ros; 87] we will denote them as SIEgroupoids, i.e.SIE = Mod{(χ • y) • y « χ, χ • χ & χ, (χ • y) • (u · ν) « (χ · u) • (y · υ)}.The variety SIE is a subvariety of the variety of binary modes which is defined only by the medial and the idempotent law. For instance, the reflectionof a point χ at a point y of the real line, i.e. χ * y := 2y — χ satisfies ailaxioms of SIE since(x * y) * y = 2y - (2y - x) = x,χ ^ χ — 2«π χ — χ j(x *y) * (u*v) = 2(2v — u) —

Journal

Demonstratio Mathematicade Gruyter

Published: Jan 1, 1999

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