States based on groups in bounded semihoops
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In this paper, we relate a bounded semihoop $ A $ to an abelian $ \ell $-group $ G $ to study $ G $-states. Firstly, the notion of Bosbach $ G $-states on bounded semihoops is presented, which reflects a certain characteristic of energy conservation. We also obtain some important conclusions of Bosbach $ G $-state as follows: $ (1) $ Let $ s $ be a Bosbach $ G $-state on a bounded semihoop $ A $ and $ H $ be a down-set subgroup of $ G $. Then $ s^{-1}(H) $ is an ideal of $ A $. $ (2) $ Every bounded dual perfect semihoop has a non-zero Bosbach $ G $-state. Secondly, we propose the notion of Rie$\check{c}$an $ G $-states on bounded semihoops and obtain some properties. We prove that every Bosbach $ G $-state is a Rie$\check{c}$an $ G $-state on a bounded semihoop $ A $ but a Rie$\check{c}$an $ G $-state may not be a Bosbach $ G $-state unless $ A $ has the DNP property. Finally, we introduce the concept of $ G $-state morphisms and obtain some characterizations of $ G $-state morphisms.
Abstract
In this paper, we relate a bounded semihoop $ A $ to an abelian $ \ell $-group $ G $ to study $ G $-states. Firstly, the notion of Bosbach $ G $-states on bounded semihoops is presented, which reflects a certain characteristic of energy conservation. We also obtain some important conclusions of Bosbach $ G $-state as follows: $ (1) $ Let $ s $ be a Bosbach $ G $-state on a bounded semihoop $ A $ and $ H $ be a down-set subgroup of $ G $. Then $ s^{-1}(H) $ is an ideal of $ A $. $ (2) $ Every bounded dual perfect semihoop has a non-zero Bosbach $ G $-state. Secondly, we propose the notion of Rie$\check{c}$an $ G $-states on bounded semihoops and obtain some properties. We prove that every Bosbach $ G $-state is a Rie$\check{c}$an $ G $-state on a bounded semihoop $ A $ but a Rie$\check{c}$an $ G $-state may not be a Bosbach $ G $-state unless $ A $ has the DNP property. Finally, we introduce the concept of $ G $-state morphisms and obtain some characterizations of $ G $-state morphisms.
참고문헌 (21)
- C. C. Chang, "Algebraic analysis of many valued logics", Transactions of the American Mathematical society, 88(2), pp.467~490 (1958)
- P. H´ajek, "Metamathematics of fuzzy logic" (1998), Kluwer Academic Publishers
- F. Esteva, "Monoidal t-norm based logic: towards a logic for left-continuous t-norms", Fuzzy Sets and Systems, 124(3), pp.271~288 (2001)
- G. J. Wang, "Non-classical mathematical logic and approximate reasoning" (2000), Science in China Press
- F. Esteva, "Hoops and fuzzy logic", Journal of Logic and Computation, 13(4), pp.532~555 (2003)
- M. Ward, "Residuated lattices", Proceedings of the National Academy of Sciences, 24(3), pp.162~164 (1938)
- R. A. Borzooei, "Local and perfect semihoops", Journal of Intelligent and Fuzzy Systems, 29(1), pp.223~234 (2015)
- H. L. Niu, "Tense operators on bounded semihoops", Pure and Applied Mathematics, 35(3), pp.325~335 (2019)
- H. L. Niu, "Ideal theory on bounded semihoops", Italian Journal of Pure and Applied Mathematics, 26, pp.911~925 (2020)
- V. Nov´ak, "Mathematical Principless of Fuzzy Logic" (1999), Kluwer Academic Publishers
- T. Hailperin, "Sentential Probability Logic" (1996), Associated University Press
- D. Mundici, "Averaging the truth-value in Lukasiewicz logic", Studia Logica, 55(1), pp.113~127 (1995)
- C. C. Lavinia, "State BL-algebras", Soft Comput, 15(4), pp.619~634 (2011)
- R. A. Borzooei, "States on EQ-algebras", Journal of Intelligent and Fuzzy Systems, 29(1), pp.209~221 (2015)
- D. Anatolij, "States on Pseudo MV-algebras", Studia Logica : An International Journal for Symbolic Logic, 68(3), pp.301~327 (2001)
- K. Michiro, "States on bounded commutative residuated lattices", Mathematica Slovaca, 64(5), pp.1093~1104 (2014)
- P. F. He, "States and internal states on semihoops", Soft Computing, 21(11), pp.2941~2957 (2017)
- P. F. He, "The existence of states based on Glivenko semihoops", Archive for Mathematical Logic, 61, pp.1145~1170 (2022)
- A. Bigard, "Groupes et Anneaux R´eticul´es" (1977), Springer-Verlag
- W. Szmielew, "Elementary properties of Abelian groups", Fundamenta Mathematicae, 41, pp.203~271 (1955)
- G. Georgescu, "Pseudo-hoops", Journal of Multiple-valued Logic and Soft Computing, 11(1-2), pp.153~184 (2014)