A Residuated Lattice of L-Fuzzy Subalgebras of a Mono-Unary Algebra

S. V. Tchoffo Foka and Marcel Tonga ()
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S. V. Tchoffo Foka: Department of Mathematics, University of Yaounde 1, P.O. Box 812, Yaounde, Cameroon
Marcel Tonga: Department of Mathematics, University of Yaounde 1, P.O. Box 812, Yaounde, Cameroon

New Mathematics and Natural Computation (NMNC), 2019, vol. 15, issue 03, 539-551

Abstract: Given a complete residuated lattice ℒ:=(L;∧,∨,⊖,↠,⊸;0,1) and a mono-unary algebra 𝒜:=(A; f), it is well known that ℒ and the residuated lattice ℱu(A,L):=(Fu(A,L);∧,∨,⊖,↠,⊸;0̲,1̲) of L-fuzzy subsets of A satisfy the same residuated lattice identities. In this paper, we show that ℒ and the residuated lattice ℱs(𝒜,L):=(Fs(𝒜,L);∧,∨,⊖,↪,↬;0̲,1̲) of L-fuzzy subalgebras of 𝒜 satisfy the same residuated lattice identities if and only if the Heyting algebra 𝒮ub(𝒜):=(Sub(𝒜);∩,∪,⇒;∅,A) of subuniverses of 𝒜 is a Boolean algebra. We also show that ℱs(𝒜,L) is a Boolean algebra (respectively, an MV-algebra) if and only if ℒ is a Boolean algebra (respectively, an MV-algebra) and 𝒮ub(𝒜) is a Boolean algebra.

Keywords: Residuated lattice; MV-algebra; Boolean algebra; mono-unary algebra; subuniverse; L-fuzzy subset; L-fuzzy subalgebra (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1142/S1793005719500315

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