Pseudo L-algebras

Document Type : Research Paper

Authors

School of Science, Xi'an Polytechnic University, Xi'an 710048, China

Abstract

We introduce generalized structures of L-algebras, called pseudo L-algebras, which are the multiplication reduct of pseudo hoops and are structures combining two L-algebras with one compatible order. We prove that every pseudo hoop gives rise to a pseudo L-algebra and every pseudo effect algebra gives rise to a pseudo L-algebra. The self-similarity is the most important property of an L-algebra $L$, which guarantees to induce a multiplication on $L$. We introduce a notion of self-similar pseudo L-algebras and prove that a self-similar pseudo L-algebra becomes an L-algebra if and only if the multiplication $\odot$ is commutative. We get some interesting results for self-similar pseudo L-algebras: (1) The negative cone $G^-$ of an $\ell$-group $G$ can be seen as a self-similar pseudo L-algebra. (2) Every self-similar pseudo L-algebra is a pseudo hoop. Next, we introduce the notion of self-similar closures of pseudo L-algebras and obtain a self-similar closure by a recursive method. Given a pseudo L-algebra $(L, \rightarrow, \rightsquigarrow ,1)$, we can generate a free semigroup $(A, \ast)$ by the set $L\setminus \{1\}$. Furthermore, we let $S(L)=A\cup\{1\}$ and define a binary operation $\odot$ on $S(L)$. Then we extend the operations $\rightarrow$ and $\rightsquigarrow$ from $L$ to $S(L)$, and prove that $(S(L), \rightarrow, 1)$ and $(S(L), \rightsquigarrow, 1)$ are two cycloids, respectively. Furthermore, under some conditions, $(S(L), \rightarrow, \rightsquigarrow, 1)$ becomes a self-similar pseudo L-algebra.  Finally, we introduce the notion of the structure group of pseudo L-algebras, and give an interesting example to show how to extend a pseudo L-algebra $L$ into the pseudo self-similar closure $S(L)$, and furthermore, derive it's structure group $G(L)$.

Keywords

References

[1] B. Bosbach, Komplementäre Halbgruppen, Axiomatik und Aritmetik, Fundamenta Mathematicae, 64 (1969), 257- 287.
[2] B. Bosbach, Komplementäre Halbgruppen, Kongruenzen und Quotienten, Fundamenta Mathematicae, 69 (1970), 1-14.
[3] B. Bosbach, Rechtskomplementare Halbgruppen, Mathematische Zeitschrift, 124 (1972), 273-288.
[4] L. C. Ciungu, Commutative pseudo BE-algebras, Iranian Journal of Fuzzy Systems, 13 (2016), 131-144.
[5] L. C. Ciungu, Results in L-algebras, Algebra Universalis, 82 (2021), 1-18.
[6] M. R. Darnel, Theory of lattice-ordered groups, Marcel Dekker Inc., New York, 1995.
[7] P. Dehornoy, Groups with a complemented presentation, Journal of Pure and Applied Algebra, 116(1-3) (1997), 115-137.
[8] P. Dehornoy, Groupes de Garside, Annales Scientifiques de ÍEcole Normale Supérieure, 35(2) (2002), 267-306.
[9] A. Dvurečenskij, Pseudo-MV algebras are intervals in ℓ-groups, Journal of the Australian Mathematical Society, 70 (2002), 427-445.
[10] A. Dvurečenskij, T. Vetterlein, Pseudo-effect algebras. I. Basic properties, International Journal of Theoretical Physics, 40(3) (2001), 685-701.
[11] P. Etingof, T. Schedler, A. Soloviev, Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Mathematical Journal, 100 (1999), 169-209.
[12] D. J. Foulis, S. Pulmannová, E. Vinceková, Lattice pseudo-effect algebras as double residuated structures, Soft Computing, 15 (2011), 2479-2488.
[13] P. Gabriel, M. Zisman, Calculus of fractions and homotopy theory, Springer, Berlin-Heidelberg-New York, 1967.
[14] G. Georgescu, A. Iorgulescu, Pseudo-MV algebras: A non-commutative extension of MV-algebras, In: Proc. 4th International Symposium of Economic Informatics, INFOREC Printing House, Bucharest, (1999), 961-968.
[15] G. Georgescu, A. Iorgulescu, Pseudo-BL algebras: A non-commutative extension of BL-algebras, In: Abstracts of the 5th International Conference FSTA 2000, Slovakia, 90-92.
[16] G. Georgescu, A. Iorgulescu, Pseudo-MV algebras, Multiple-Valued Logic, 6 (2001), 95-135.
[17] G. Georgescu, A. Iorgulescu, Pseudo-BCK algebras: An extension of BCK-algebras, In: Proc. DMTCS01: Combinatorics, Computability and Logic, Springer, London, (2001), 97-114.
[18] G. Georgescu, L. Leuştean, V. Preoteasa, Pseudo-hoops, Journal of Multiple-Valued Logic and Soft Computing, 11 (2005), 153-184.
[19] L. Herman, E. L. Marsden, R. Piziak, Implication connectives in orthomodular lattices, Notre Dame Journal of Formal Logic, 16 (1975), 305-328.
[20] A. Iorgulescu, Pseudo-Iséki algebras. Connection with pseudo-BL algebras, Journal of Multiple-Valued Logic and Soft Computing, 11(3-4) (2005), 263-308.
[21] A. Iorgulescu, Classes of pseudo-BCK algebras-Part I, Journal of Multiple-Valued Logic and Soft Computing, 12(1-2) (2006), 71-130.
[22] A. Iorgulescu, Classes of pseudo-BCK algebras-Part II, Journal of Multiple-Valued Logic and Soft Computing, 12(5-6) (2006), 575-629.
[23] A. Iorgulescu, Classes of examples of pseudo-MV algebras, pseudo-BL algebras and divisible bounded noncommutative residuated lattices, Soft Computing, 14(4) (2010), 313-327.
[24] I. Leustean, Non-commutative Łukasiewicz propositional logic, Archive for Mathematical Logic, 45 (2006), 191-213.
[25] W. Rump, A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation, Advances in Mathematics, 193 (2005), 40-55.
[26] W. Rump, Braces, radical rings, and the quantum Yang-Baxter equation, Journal of Algebra, 307 (2007), 153-170.
[27] W. Rump, L-algebras, self-similarity, and ℓ-groups, Journal of Algebra, 320 (2008), 2328-2348.
[28] W. Rump, Right ℓ-groups, geometric Garside groups, and solutions of the quantum Yang-Baxter equation, Journal of Algebra, 439 (2015), 470-510.
[29] W. Rump, The structure group of an L-algebra is torsion-free, Group Theory, 2 (2017), 309-324.
[30] W. Rump, Von Neumann algebras, L-algebras, Baer*-monoids, and Garside groups, Forum Mathematicum, 3 (2018), 973-995.
[31] W. Rump, The structure group of a generalized orthomodular lattice, Studia Logica, 106 (2018), 85-100.
[32] W. Rump, Symmetric quantum sets and L-algebras, International Mathematics Research Notices, 1 (2020), 1-41.
[33] W. Rump, Y. C. Yang, Intervals in ℓ-groups as L-algebras, Algebra University, 67 (2012), 121-130.
[34] T. Traczyk, On the structure of BCK-algebras with zx·yx = zy ·xy, Mathematica Japonica, 33(2) (1988), 319-324.
[35] Y. Wu, J. Wang, Y. C. Yang, Lattice ordered effect algebras and L-algebras, Fuzzy Sets and Systems, 369 (2019), 103-113.
Iranian Journal of Fuzzy Systems
Volume 19, Issue 6
November and December 2022
Pages 61-73

Files

Share

How to cite

Statistics