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**On locally internal monotonic operations.**
*(English)*
Zbl 1022.03038

Summary: This paper deals with monotonic binary operations \(F: [0,1]^2 \to[0,1]\) with the property (called locally internal property) that the value of any point \((x,y)\) is always one of its arguments \(x,y\). After stating a theorem that characterizes this kind of operations, some special cases are studied in detail by considering additional properties of the operations: commutativity, existence of a neutral element and associativity. In case of locally internal, associative monotonic operations with neutral element, a characterization theorem gives an improvement of a well-known theorem of Czogala and Drewniak on idempotent, associative and increasing operations with neutral element, as well as an improvement of a characterization theorem for left (and right) continuous, idempotent uninorms.

### MSC:

03E72 | Theory of fuzzy sets, etc. |

### Keywords:

binary operation; monotonicity; commutativity; associativity; neutral element; idempotent uninorm
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\textit{J. Martín} et al., Fuzzy Sets Syst. 137, No. 1, 27--42 (2003; Zbl 1022.03038)

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### References:

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