Abstract
The idea of residuation goes back to Dedekind, who introduced it into ideal theory. Residuation now plays an important role in several fields of mathematics, especially commutative ring theory. Let R be a commutative ring with identity. The residual of an ideal B with respect to an ideal A is the ideal A: B = {x ∈ R | xB ⊆ A. Here A: B is the largest ideal X of R with the property that BX ⊆ A. Residuation may be defined in other algebraic structures and may be defined independently of multiplication.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
D.D. Anderson, Distributive Noether lattices, Michigan Math. J. 22 (1975), 109–115.
—, Fake rings, fake modules, and duality, J. Algebra 47 (1977), 425–432.
—, and E.W. Johnson, Ideal theory in commutative semigroups, Semigroup Forum 30 (1984), 127–158.
T.S. Blyth and M.F. Janowitz, “Residuation Theory,” Pergamon Press, Oxford, 1972.
M. Raquel P. da Costa Reis, Partial residuals in groupoid-lattices-I, Math. Japonica 25 (1980), 19–26.
P. da Costa Reis —, Partial residuals in groupoid-lattices-II, Math. Japonica 25 (1980), 607–618.
P. da Costa Reis —, Partial residuals in groupoid-lattices-III, Math. Japonica 26 (1981), 467–474.
R.P. Dilworth, Abstract residuation over lattices, Bull. Amer. Math. Soc. 44 (1938), 262–267. Reprinted in Chapter 6 of this volume.
—, Non-commutative arithmetic, Duke Math. J. 5 (1939), 270–280. Reprinted in Chapter 6 of this volume.
—, Non-commutative residuated lattices, Trans. Amer. Math. Soc. 46 (1939), 426–444. Reprinted in Chapter 6 of this volume.
—, Abstract commutative ideal theory, Pacific J. Math. 12 (1962), 481–498. Reprinted in Chapter 6 of this volume.
M.L. Dubriel-Jacotin, L. Lesieur, and R. Croisot, “ Leçons sur la Théorie des Treillis des Structures Algébrique Ordonnés et des Treillis Géométriques,” Gauthier-Villars, Paris, 1953.
I. Fleischer, A lattice theoretic look at some ring theoretic radicals, preprint.
L. Fuchs, “Partially Ordered Algebraic Systems,” Pergamon Press, Oxford, 1963.
W.H. Rowan, Tertiary decomposition in lattice modules, Algebra Universalis 22 (1986), 27–49.
O. Steinfeld, On groupoid-lattices, in “Contributions to General Algebra, Proceedings of the Klagenfurt conference,” 1978, pp. 357-372.
M. Ward, Residuation in structures over which a multiplication is defined, Duke Math. J. 3 (1937), 627–636.
—, Structure residuation, Ann. of Math. (2) 39 (1938), 558–568.
—, Residuated distributive lattices, Duke Math. J. 6 (1940), 641–651.
—, and R.P. Dilworth, Residuated lattices, Proc. Nat. Acad. Sci. 24 (1938), 162–164.
R.P. Dilworth —, Residuated lattices, Trans. Amer. Math. Soc. 35 (1939), 335–354. Reprinted in Chapter 6 of this volume.
R.P. Dilworth —, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608.
R.P. Dilworth —, Evaluations over residuated structures, Ann. of Math. (2) 40 (1939), 328–338.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer Science+Business Media New York
About this chapter
Cite this chapter
Anderson, D.D. (1990). Dilworth’s Early Papers on Residuated and Multiplicative Lattices. In: Bogart, K.P., Freese, R., Kung, J.P.S. (eds) The Dilworth Theorems. Contemporary Mathematicians. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-3558-8_36
Download citation
DOI: https://doi.org/10.1007/978-1-4899-3558-8_36
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4899-3560-1
Online ISBN: 978-1-4899-3558-8
eBook Packages: Springer Book Archive