A Bijection between Lattice-Valued Filters and Lattice-Valued Congruences in Residuated Lattices

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Abstract
The aim of this paper is to study relations between lattice-valued filters and lattice-valued congruences in residuated lattices. We introduce a new definition of congruences which just depends on the meet∧and the residuum→. Then it is shown that each of these congruences is automatically a universal-algebra-congruence. Also, lattice-valued filters and lattice-valued congruences are studied, and it is shown that there is a one-to-one correspondence between the set of all (lattice-valued) filters and the set of all (lattice-valued) congruences.
Funding Information
  • Education Department of Shaanxi Province (2013JK1139, 61202163, 61240035, 2012011015-1)
  • National Natural Science Foundation of China (2013JK1139, 61202163, 61240035, 2012011015-1)
  • National Natural Science Foundation of China (2013JK1139, 61202163, 61240035, 2012011015-1)
  • Natural Science Foundation of Shaanxi Province (2013JK1139, 61202163, 61240035, 2012011015-1)

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