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We investigate the class of bisymmetric and quasitrivial binary operations on a given set $X$ and provide various characterizations of this class as well as the subclass of bisymmetric, quasitrivial, and order-preserving binary operations. We also determine explicitly the sizes of these classes when the set $X$ is finite., Comment: arXiv admin note: text overlap with arXiv:1709.09162

The aim of the paper is to investigate the solutions of special inhomogeneous linear functional equations using spectral analysis in a translation invariant closed linear subspace of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The application of spectral analysis in some related varieties is a new and important trend in the theory of functional equations; especially they have successful applications in the case of homogeneous linear functional equations. The foundations of the theory can be found in Kiss and Varga (Aequat Math 88(1):151–162, 2014) and Kiss and Laczkovich (Aequat Math 89(2):301–328, 2015). We are going to adopt the main theoretical tools to solve some inhomogeneous problems due to Koclȩga-Kulpa and Szostok (Ann Math Sylesianae 22:27–40, 2008), see also Koclȩga-Kulpa and Szostok (Georgian Math J 16:725–736, 2009; Acta Math Hung 130(4):340–348, 2011). They are motivated by quadrature rules of approximate integration.

We introduce the concept of quasi-Lovasz extension as being a mapping $${f\colon I^n\to\mathbb{R}}$$ defined on a nonempty real interval I containing the origin and which can be factorized as f(x 1, . . . , x n ) = L(φ(x 1), . . . , φ(x n )), where L is the Lovasz extension of a pseudo-Boolean function $${\psi\colon \{0, 1\}^n \to \mathbb{R}}$$ (i.e., the function $${L\colon \mathbb{R}^n \to \mathbb{R}}$$ whose restriction to each simplex of the standard triangulation of [0, 1] n is the unique affine function which agrees with ψ at the vertices of this simplex) and $${\varphi\colon I \to \mathbb{R}}$$ is a nondecreasing function vanishing at the origin. These functions appear naturally within the scope of decision making under uncertainty since they subsume overall preference functionals associated with discrete Choquet integrals whose variables are transformed by a given utility function. To axiomatize the class of quasi-Lovasz extensions, we propose generalizations of properties used to characterize Lovasz extensions, including a comonotonic version of modularity and a natural relaxation of homogeneity. A variant of the latter property enables us to axiomatize also the class of symmetric quasi-Lovasz extensions, which are compositions of symmetric Lovasz extensions with 1-place nondecreasing odd functions.

Kolmogoroff and Nagumo proved that the quasi-arithmetic means correspond exactly to the decomposable sequences of continuous, symmetric, strictly increasing in each variable and reflexive functions. We replace decomposability and symmetry in this characterization by a generalization of the decomposability.