×

The structure of residuated lattices. (English) Zbl 1048.06010

Summary: A residuated lattice is an ordered algebraic structure \[ \mathbf L=\langle L,\vee,\wedge,\cdot,e,\setminus,/\rangle \] such that \(\langle L,\vee,\wedge\rangle\) is a lattice, \(\langle L,\cdot,e\rangle\) is a monoid, and \(\setminus\) and \(/\) are binary operations for which the equivalences \[ a\cdot b\leq c \Leftrightarrow a \leq c/b \Leftrightarrow b \leq a\setminus c \] hold for all \(a,b,c \in L\). It is helpful to think of the last two operations as left and right division and thus the equivalences can be seen as “dividing” on the right by \(b\) and “dividing” on the left by \(a\). The class of all residuated lattices is denoted by \(\mathcal {RL}\).
The study of such objects originated in the context of the theory of ring ideals in the 1930s. The collection of all two-sided ideals of a ring forms a lattice upon which one can impose a natural monoid structure making this object into a residuated lattice. Such ideas were investigated by Morgan Ward and R. P. Dilworth in a series of important papers and also by Krull. Since that time, there has been substantial research regarding some specific classes of residuated structures, but we believe that this is the first time that a general structural theory has been established for the class \(\mathcal{RL}\) as a whole. In particular, we develop the notion of a normal subalgebra and show that \(\mathcal{RL}\) is an “ideal variety” in the sense that it is an equational class in which congruences correspond to “normal” subalgebras in the same way that ring congruences correspond to ring ideals. As an application of the general theory, we produce an equational basis for the important subvariety \(\mathcal{RL}^{\mathcal C}\) that is generated by all residuated chains. In the process, we find that this subclass has some remarkable structural properties that we believe could lead to some important decomposition theorems for its finite members.

MSC:

06F25 Ordered rings, algebras, modules
06B05 Structure theory of lattices
06B20 Varieties of lattices
06B10 Lattice ideals, congruence relations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1007/978-94-009-2871-8
[2] Balbes R., Distributive Lattices (1974)
[3] Benado M., C. R. Acad. Sci., Paris 228 pp 529–
[4] Birkhoff G., Lattice Theory 25 (1967) · Zbl 0153.02501
[5] Blyth T. S., Residuation Theory 102 (1972) · Zbl 0301.06001
[6] Burris S., A Course in Univeral Algebra 78, in: Graduate Texts in Mathematics (1981)
[7] Cignoli R. L. O., Algebraic Foundations of Many-Valued Reasoning 7, in: Trends in Logic (1999)
[8] DOI: 10.1090/S0002-9947-1961-0121405-2
[9] Conrad P., Czech. Math. J. 15 pp 101–
[10] DOI: 10.1007/BF00370141 · Zbl 0854.06011
[11] Davey B. A., Introduction to Lattices and Order (1990) · Zbl 0701.06001
[12] DOI: 10.1090/S0002-9904-1938-06736-5 · Zbl 0018.34104
[13] DOI: 10.1090/S0002-9947-1939-0000230-5 · JFM 65.0084.02
[14] Fuchs L., Partially Ordered Algebraic Systems 28 (1963) · Zbl 0137.02001
[15] DOI: 10.1007/978-3-642-67678-9
[16] Gilmer R. W., Multiplicative Ideal Theory 12, in: Queen’s Papers in Pure and Applied Mathematics (1968)
[17] DOI: 10.1142/3811
[18] Glass A. M. W., Lattice-Ordered Groups, in: Mathematics and Its Applications (1989) · Zbl 0708.06015
[19] Grätzer G., Lattice Theory: First Concepts and Distributive Lattices (1971) · Zbl 0232.06001
[20] DOI: 10.1007/978-3-0348-7633-9
[21] DOI: 10.1007/BF01191491 · Zbl 0547.08001
[22] Hájek P., Metamathematics of Fuzzy Logic 4, in: Trends in Logic (1998)
[23] DOI: 10.1090/S0002-9947-1994-1211409-2
[24] Hart J. B., Int. J. Algebra Comput.
[25] Holland W. C., Michigan Math. J. 10 pp 399–
[26] DOI: 10.1080/00927879408824863 · Zbl 0802.06018
[27] Jónsson B., Math. Scand. 21 pp 110– · Zbl 0167.28401
[28] Krull W., Sitzungsberichte der physikalischmedicinischen Societät. zu Erlangen 56 pp 47–
[29] Larsen M. D., Pure and Applied Mathematics 43, in: Multiplicative Theory of Ideals (1971)
[30] McCarthy P. J., Acta. Sci. Math. (Szeged) 27 pp 63–
[31] McKenzie R. N., Algebras, Lattices and Varieties (1987)
[32] DOI: 10.1007/978-1-4615-5217-8
[33] DOI: 10.1007/BF02039527 · Zbl 0031.28002
[34] DOI: 10.1080/00927878908823857 · Zbl 0689.06015
[35] DOI: 10.1007/978-94-009-2283-9_12
[36] DOI: 10.1090/S0002-9947-1946-0015126-6
[37] DOI: 10.1007/BF01190765 · Zbl 0819.06009
[38] Ursini A., Boll. Un. Mat. Ital. 6 pp 90–
[39] DOI: 10.1215/S0012-7094-37-00351-X · Zbl 0018.19903
[40] DOI: 10.2307/1968634 · Zbl 0019.28902
[41] DOI: 10.1073/pnas.24.3.162 · Zbl 0018.29003
[42] DOI: 10.1090/S0002-9947-1939-1501995-3 · JFM 65.0084.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.