CONTENTS Introduction.......................................................................................................... 5 II. Quasi-transitive algebraic objects....................................................................... 12 III. Rank of the quasi-transitivity of algebraic objects........................................... 22 IV. Commutative algebraic objects.......................................................................... 25 V. Regular algebraic objects..................................................................................... 31 VI. Particular algebraic objects.................................................................................. 38 VII. Reduced algebraic objects.................................................................................. 49 VIII. Translation equation and algebraic objects over groupoids....................... 52 References................................................................................................................... 61
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There are many inequalities which in the class of continuous functions are equivalent to convexity (for example the Jensen inequality and the Hermite-Hadamard inequalities). We show that this is not a coincidence: every nontrivial linear inequality which is valid for all convex functions is valid only for convex functions.
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There are many types of midconvexities, for example Jensen convexity, t-convexity, (s,t)-convexity. We provide a uniform framework for all the above mentioned midconvexities by considering a generalized middle-point map on an abstract space X. We show that we can define and study the basic convexity properties in this setting.
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