Abstract
The aim of this paper is to investigate several operators on L-algebras. At first, closure (interior) operators on L-algebras are defined and some properties of them are obtained. Then, existential operators and universal operators on L-algebras are studied, a one-to-one correspondence between the set of all quantifier operators and the set of all relative complete subalgebras of CKL-algebras is constructed. Furthermore, very true operators on L-algebras are investigated and by giving a very true ideal of a very true L-algebra, quotient structures on very true L-algebras are established.
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References
1.
Belluce
L.P.
,
Grigolia
R.
and
Lettieri
A.
, Representations of monadic MV-algebras , Stud Log 81 (2005 ), 123 –144 .
2.
Bezhanishvili
G.
and
Harding
J.
, Functional monadic Heyting algebras , Algebra Univ 48 (2002 ), 1 –10 .
3.
Chajda
I.
and
Kolařák,
M.
, Very true operators in effect algebras , Soft Comput 16 (2012 ), 1213 –1218 .
4.
Di Nola
A.
,
Grigolia,
R.
, On monadic MV-algebras , Ann PureAppl Log 128 (2004 ), 125 –139 .
5.
Galatos
N.
,
Jipsen
P.
,
Kowalski
T.
,
Ono
H.
Residuated lattices: An algebraic glimpse at substructural logics. Elsevier, New York, 2007.
6.
Georgescu
G.
,
Iorgulescu
A.
and
Leustean
I.
, Monadic and closureMV-algebras , Multi Val Log 3 (1998 ), 235 –257 .
7.
Halmos
P.
, Algebraic logic I: monadic Boolean algebras , Compos Math 12 (1955 ), 217 –249 .
8.
Hájek
P.
, On very true , Fuzzy Sets Syst 124 (2001 ), 329 –333 .
9.
Leustean
I.
, Non-commutative Lukasiewicz propositional logic , Arch Math Logic 45 (2006 ), 191 –213 .
10.
Rachunek
J.
and
Šalounová,
D.
, Truth values on generalizations of some commutative fuzzy structures , Fuzzy Sets Syst 157 (2006 ), 3159 –3168 .
11.
Rutledge
J.D.
A preliminary investigation of the infinitely many-valued predicate calculus, Ph.D. Thesis, Cornell University, 1959.
12.
Rump
W.
, A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equations , Adv Math 193 (2005 ), 40 –55 .
13.
Rump
W.
, L -algebras, self-similarity, and L -groupsJ Algebra 320 (2008 ), 2328 –2348 .
14.
Rump
W.
, L -algebras and three main non-classical logicsAnnals of Pure and Applied Logic 173 (2022 ), 103121 .
15.
Rump
W.
, A general Glivenko theorem , Algebra Univers 61 (2009 ), 455 –473 .
16.
Rump
W.
, L -algebras in logic, algebra, geometry, and topologySoft Comput 24 (2020 ), 3077 –3085 .
17.
Rump
W.
, The L -algebra of Hurwitz primes , J Number Theory 190 (2018 ), 394 –413 .
18.
Rump
W.
, Right-groups, geometric Garside groups, and solutions of the quantum Yang-Baxter equation , J Algebra 439 (2015 ), 470 –510 .
19.
Rump
W.
, Von Neumann algebras, L -algebras, Baer*-monoids, and Garside groups , Forum Math 3 (2018 ), 973 –995 .
20.
Rump
W.
, The structure group of a generalized orthomodular lattice , Stud Logica 106 (2018 ), 85 –100 .
21.
Solovyov
S.A.
, On monadic quantale algebras: basic properties andrepresentation theorems , Discuss Math Gen Algebra Appl 30 (2010 ), 91 –118 .
22.
Burris
S.
,
Sankappanavar
H.P.
A course in universal algebra. (1981), Springer, Berlin.
23.
Wang
J.T.
,
Xin
X.L.
and
Jun
Y.B.
, Very true operators on equality algebras , J Comput Anal Appl 24 (2018 ), 507 –521 .
24.
Wu
Y.L.
,
Wang
J.
and
Yang
Y.C.
, Lattice-ordered effect algebras and L -algebras , Fuzzy Sets Syst 369 (2019 ), 103 –113 .
25.
Yang
Y.C.
and
Rump
W.
, Pseudo-MV algebras as L -algebras , J. Mult-valued Logic Soft Comput 19 (2012 ), 621 –632 .
26.
Zadeh
L.
, A fuzzy-set-theoretical interpretation of linguistic hedges , J Cybern 2 (1972 ), 4 –34 .
27.
Zhan
Q.Y.
,
Wang
J.
,
Luan
W.
and
Yang
Y.C.
, Fuzziness in L -algebras , Fuzzy Sets and Syst 425 (2021 ), 157 –168 .