ISSN:
1420-8911
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract p(x, y, z) is aternary deduction (TD) term function on an algebra A if, for alla, b ε A, p(a, b,z) ≡ z (modΘ (a, b)), and {p(a, b, z): z εA} is a transversal of the set of equivalence classes of the principal congruence θ(a, b). p iscommutative ifp(a, b, z) and p(a', b', z) define the same transversal whenever0(a, b)=0(a', b'). p isregular ifΘ(p(x, y, 1), 1)=0(x, y) for some constant term 1. The TD term generalizes the (affine) ternary discriminator and is used to investigate the logical structure of nonsemisimple varieties with equationally definable principal congruences (EDPC). Some of the results obtained: The following are equivalent for any variety: (1)V has a TD term; (2)V has EDPC and a certain strong form of the congruence-extension property. IfV is semisimple and congruence-permutable, (1) and (2) are equivalent to (3)V is an affine discriminator variety. Afixedpoint ternary discriminator on a set is defined by the conditions:p(x, x,z)=z and, ifx ≠ y, p(x, y,z)=d whered is some fixed element; afixedpoint discriminator variety is defined in analogy to affine discriminator variety. The commutative TD term generalizes the fixedpoint ternary discriminator. The following are equivalent for any semisimple variety: (4)V has a commutative TD term; (5)V is a fixedpoint discriminator variety. IfV is semisimple, congruence-permutable, and has a constant term, (4) and (5) are equivalent to (3); ifV has a second constant term distinct from the first in all nontrivial members ofV then all five conditions are equivalent to (6)V has a commutative, regular TD term. Ahoop is a commutative residuated monoid.Hoops with dual normal operators are defined in analogy with normal Boolean algebras with operators. The main result: A variety of hoops with dual normal operators has a commutative, regular TD term iff it has EDPC iff it has first-order definable principal congruences.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01195727
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