De Miguel, L.; Bustince, H.; De Baets, B. Convolution lattices. (English) Zbl 1397.06004 Fuzzy Sets Syst. 335, 67-93 (2018). Summary: We propose two convolution operations on the set of functions between two bounded lattices and investigate the algebraic structure they constitute, in particular the lattice laws they satisfy. Each of these laws requires the restriction to a specific subset of functions, such as normal, idempotent or convex functions. Combining all individual results, we identify the maximal subsets of functions resulting in a bounded lattice, and show this result to be equivalent to the distributivity of the lattice acting as domain of the functions. Furthermore, these lattices turn out to be distributive as well. Additionally, we show that for the larger subset of idempotent functions, although not satisfying the absorption laws, the convolution operations satisfy the Birkhoff equation. Cited in 8 Documents MSC: 06B05 Structure theory of lattices 06D05 Structure and representation theory of distributive lattices 06D72 Fuzzy lattices (soft algebras) and related topics Keywords:algebra; convolution operations; lattice PDFBibTeX XMLCite \textit{L. De Miguel} et al., Fuzzy Sets Syst. 335, 67--93 (2018; Zbl 1397.06004) Full Text: DOI Link References: [1] Myers, D. G., Digital signal processing: efficient convolution and Fourier transform techniques, (1990), Prentice-Hall New York (N.Y.) [2] Derighetti, A., Convolution operators on groups, (2011), Springer Berlin, Heidelberg [3] Smith, S. W., The scientist and Engineer’s guide to digital signal processing, 123-140, (1997), California Technical Publishing, Ch. 7 - Properties of convolution [4] Serra, J., Image analysis and mathematical morphology, (1983), Academic Press, Inc. Orlando, FL, USA [5] De Baets, B.; Kerre, E.; Gupta, M., The fundamentals of fuzzy mathematical morphology. part 1: basic concepts, Int. J. Gen. Syst., 23, 2, 155-171, (1995) [6] De Baets, B.; Kerre, E.; Gupta, M., The fundamentals of fuzzy mathematical morphology. part 2: idempotence, convexity and decomposition, Int. J. Gen. Syst., 23, 4, 307-322, (1995) [7] De Baets, B., Uncertainty analysis in engineering and sciences: fuzzy logic, statistics, and neural network approach, 53-67, (1998), Springer US Boston, MA, Ch. 4 - Fuzzy morphology: a logical approach [8] Bloch, I.; Maître, H., Fuzzy mathematical morphologies: a comparative study, Pattern Recognit., 28, 9, 1341-1387, (1995) [9] Heijmans, H., Theoretical aspects of gray-level morphology, IEEE Trans. Pattern Anal. Mach. Intell., 13, 568-582, (1991) [10] Heijmans, H.; Ronse, C., The algebraic basis of mathematical morphology I. dilations and erosions, Comput. Vis. Graph. Image Process., 50, 3, 245-295, (1990) [11] Ronse, C., Why mathematical morphology needs complete lattices, Signal Process., 21, 2, 129-154, (1990) [12] Zadeh, L., The concept of a linguistic variable and its application to approximate reasoning - I, Inf. Sci., 8, 3, 199-249, (1975) [13] Kerre, E., A tribute to Zadeh’s extension principle, Sci. Iran., 18, 3, 593-595, (2011) [14] Moore, R.; Lodwick, W., Interval analysis and fuzzy set theory, Fuzzy Sets Syst., 135, 1, 5-9, (2003) [15] Dubois, D.; Prade, H., Possibility theory. an approach to computerized processing of uncertainty, (1988), Springer US [16] Dubois, D.; Prade, H., Additions of interactive fuzzy numbers, IEEE Trans. Autom. Control, 26, 4, 926-936, (1981) [17] Scheerlinck, K.; Vernieuwe, H.; Verhoest, N.; De Baets, B., Practical computing with interactive fuzzy variables, Appl. Soft Comput., 22, 518-527, (2014) [18] de Cooman, G., Fuzzy set theory and advanced mathematical applications, 89-133, (1995), Springer US Boston, MA, Ch. 4 - Towards a possibilistic logic [19] Van Schooten, A., Ontwerp en implementatie Van een model voor de representatie en manipulatie Van onzekerheid en imprecisie in databanken en expertsystemen, (1989), Universiteit Gent, Ph.D. thesis [20] De Tré, G.; De Baets, B., Aggregating constraint satisfaction degrees expressed by possibilistic truth values, IEEE Trans. Fuzzy Syst., 11, 3, 361-368, (2003) [21] Mizumoto, M.; Tanaka, K., Some properties of fuzzy sets of type 2, Inf. Control, 31, 4, 312-340, (1976) [22] Mizumoto, M.; Tanaka, K., Fuzzy sets and type 2 under algebraic product and algebraic sum, Fuzzy Sets Syst., 5, 3, 277-290, (1981) [23] Hernández, P.; Cubillo, S.; Torres-Blanc, C., On t-norms for type-2 fuzzy sets, IEEE Trans. Fuzzy Syst., 23, 4, 1155-1163, (2015) [24] Hernández, P.; Cubillo, S.; Torres-Blanc, C., Negations on type-2-fuzzy sets, Fuzzy Sets Syst., 252, 111-124, (2014) [25] Takáč, Z., Aggregation of fuzzy truth values, Inf. Sci., 271, 1-13, (2014) [26] Walker, C. L.; Walker, E. A., The algebra of fuzzy truth values, Fuzzy Sets Syst., 149, 2, 309-347, (2005) [27] Harding, J.; Walker, C.; Walker, E., Lattices of convex normal functions, Fuzzy Sets Syst., 159, 9, 1061-1071, (2008) [28] Walker, C. L.; Walker, E. A., Some general comments on fuzzy sets of type-2, Int. J. Intell. Syst., 24, 1, 62-75, (2009) [29] Harding, J.; Walker, E. A.; Walker, C. L., The truth value algebra of type-2 fuzzy sets, (2016), CRC Press, Taylor & Francis Group [30] Walker, C. L.; Walker, E. A., A family of finite De Morgan and Kleene algebras, Int. J. Uncertain. Fuzziness Knowl.-Based Syst., 20, 631-654, (2012) [31] Walker, C. L.; Walker, E. A., Type-2 operations on finite chains, Fuzzy Sets Syst., 236, 33-49, (2014) [32] Grätzer, G., Lattice theory: foundation, (2011), Birkhäuser [33] Birkhoff, G., Lattice theory, Colloq. Publ. - Am. Math. Soc., vol. 25, (1967), American Mathematical Society [34] Davey, B. A.; Priestley, H. A., Introduction to lattices and order, (1990), Cambridge University Press [35] Galatos, N.; Jipsen, P.; Kowalski, T.; Ono, H., Residuated lattices: an algebraic glimpse at substructural logics, (2007), Elsevier 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.