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K. Denecke, S. Wismath (2000)
Hyperidentities and Clones
(1999)
D e n e c k e , Sets of hypersubstitutions and set-solid varieties, in: Algebras and Combinatorics
S. Wismath (2001)
Fundamental-M-Solid and Fundamental-M-Closed VarietiesSoutheast Asian Bulletin of Mathematics, 25
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) —
Demonstratio Mathematica – de Gruyter
Published: Jan 1, 1999
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