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Conference Proceedings Proceedings MFOI-2017 Conference on Mathematical Foundations of Informatics Institute of Mathematics and Computer Science November 9-11, 2017, Chisinau, Moldova CZU 51+004(082) C 65 Copyright © Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, 2017. All rights reserved. INSTITUTE OF MATHEMATICS AND COMPUTER SCIENCE 5, Academiei street, Chisinau, Republic of Moldova, MD 2028 Tel: (373 22) 72-59-82, Fax: (373 22) 73-80-27, E-mail: [email protected] WEB address: http://www.math.md Editors: Prof. S.Cojocaru, Prof. C.Gaindric, Ph.D. I.Drugus. Authors are fully responsible for the content of their papers. Descrierea CIP a Camerei Naţionale a Cărţii Conference on Mathematical Foundations of Informatics, November 9‐11, 2017, Chisinau, Moldova : Proceedings MFOI‐2017 / Inst. of Mathematics and Computer Science, Acad. of Sciences of Moldova ; ed.: S. Cojocaru [et al.]. – [Chişinău] : Institute of Mathematics and Computer Science, 2017 (Tipogr. "Valinex" SRL). – 182 p. : fig., tab. Rez.: lb. engl. – Referinţe bibliogr. la sfârşitul art. şi în subsol. – 60 ex. ISBN 978‐9975‐4237‐6‐2. 51+004(082) ISBN 978‐9975‐4237‐6‐2 Proceedings of the Conference on Mathematical Foundations of Informatics MFOI2017, November 9-11, 2017, Chisinau, Republic of Moldova Results Regarding Cardinalities in FSM Andrei Alexandru, Gabriel Ciobanu Abstract In this paper we present the consistency of various results regarding cardinalities and Dedekind-finiteness in a newly devel- oped framework called Finitely Supported Mathematics. Keywords: cardinality, Dedekind-Finiteness, FSM. 1 Finitely Supported Mathematics Finitely Supported Mathematics (FSM) is a recently developed frame- work used for managing infinite structures by involving a finite number of characteristics. More exactly, in FSM we associate to each object a finite family of elements characterizing it, which is called its “finite support". The properties of each object are well described only by analyzing the properties of its ”finite support". Finitely Supported Mathematics has connections with the permuta- tion models of Zermelo-Fraenkel set theory with atoms, with Fraenkel- Mostowski (FM) axiomatic set theory, with the theory of nominal sets (which is a Zermelo-Fraenkel (ZF) alternative to axiomatic FM set theory), and with the theory of admissible sets (because the sets con- structed in FSM are hereditary finitely supported sets). More precisely, the theory of nominal sets originally developed over a fixed countable set of atoms is extended to a theory of invariant sets over a fixed infinite (possible non-countable) set of atoms. An invariant set (X; ·) is actu- ally a classical ZF set X equipped with an action · on X of the group of permutations of a (possible non-countable) fixed ZF set A (called the set of atoms), having the additional property that any element x 2 X c 2017 by Andrei Alexandru, Gabriel Ciobanu 3 Andrei Alexandru, Gabriel Ciobanu is finitely supported. In a pair (X; ·) formed by a ZF set X and a group action · on X of the group of all permutations of atoms, an arbitrary element x 2 X is finitely supported if there exists a finite family S of atoms such that any permutation of atoms that fixes S pointwise also leaves x invariant under the group action ·. A relation (or, par- ticularly, a function) between two invariant sets is finitely supported if it is finitely supported as a subset of the Cartesian product of those two invariant sets. The theory of invariant sets allows us to define in- variant algebraic structures (as invariant sets endowed with invariant algebraic laws) which are used in order to construct FSM [1]. Con- cretely, FSM represents a reformulation of the ZF algebra obtained by replacing ‘(infinite) structure' with `invariant/finitely supported struc- ture'. All the objects defined in FSM must be constructed according to the finite support principle. As proved in [1], not every ZF result can be directly rephrased in FSM. As a consequence, we cannot obtain a property in FSM only by involving a ZF result without an appropri- ate proof reformulated according to the finite support requirement (the proof should be itself internally consistent in FSM, and not retrieved from ZF). The algorithmic techniques for translating ZF results into FSM are described in [1]. By applying them we are able to present some results regarding cardinality and Dedekind-finiteness, in the world of finitely supported structures. 2 Cardinalities in FSM In ZF, the trichotomy law for cardinalities states that any two cardinals are comparable. More precisely, the first ZF trichotomy principle states that for every two non-empty sets there is an one-to-one mapping from one into the other, while the second ZF trichotomy principle states that for every two non-empty sets there is a mapping of one onto the other. In ZF these principles are both equivalent with the axiom of choice. However, as we said before, the ZF relationship results (particularly those between choice and trichotomy) does not necessarily remain valid in FSM. Both trichotomy principles for cardinalities fail in FSM. 4 Results Regarding Cardinalities in FSM Theorem 2.1. 1. There may exist two non-empty invariant sets (par- ticularly the set A of atoms and the set N of all positive integers) such that there is no finitely supported one-to-one mapping from one into the other. 2. There may exist two non-empty invariant sets (particularly the set A of atoms and the set N of all positive integers) such that there is no finitely supported mapping of one onto the other. In ZF set theory with the axiom of choice it is known that Tarski's theorem about choice holds, meaning that for any infinite set X there exists a bijection between X and X × X. This result fails in FSM. The existence of right inverses for surjective functions also fail in FSM. Theorem 2.2. 1. There exists an infinite invariant set X (particularly the set A of atoms) such that there is no finitely supported surjection f : X ! X × X. 2. There exists a finitely supported surjective function between two invariant sets which has no finitely supported right inverse. Schr¨oder-Bernsteintheorem which emphasizes the antisymmetry of cardinal ordering remains consistent in FSM. While Schr¨oder-Bernstein theorem is consistent in FSM, its dual is no longer valid in this new framework. Theorem 2.3. 1. Let B and C be two invariant sets such that there exist a finitely supported one-to-one mapping f : B ! C and a finitely supported one-to-one mapping g : C ! B. Then there exists a finitely supported bijective mapping h : B ! C. 2. There are two invariant sets B and C such that there exists a finitely supported onto mapping f : B ! C and a finitely supported onto mapping g : C ! B, but there does not exist a finitely supported bijective mapping h : B ! C. 3 Dedekind Finiteness in FSM In ZF, a set is called Dedekind-infinite provided that there exists an in- jection into one of its proper subsets. Otherwise, it is called Dedekind- 5 Andrei Alexandru, Gabriel Ciobanu finite. In ZF with choice the concepts ‘finite’ and ‘Dedekind-finite’ coincide. In FSM there exist infinite sets that are Dedekind-finite. Theorem 3.1. The concepts of finiteness and Dedekind-finiteness are different in FSM, meaning that there exists an infinite invariant set (particularly the set A of atoms) which has no finitely supported injec- tion into any of its finitely supported proper subsets. Other examples of infinite invariant sets which are Dedekind-finite in FSM are the set of all finite subsets of A, the set of all finitely supported (i.e. either finite or cofinite) subsets of A, and the set of all finite injective tuples of atoms. In ZF with the axiom of choice, a set X is finite if and only if every surjective function from X onto itself is one-to-one. Such a character- ization of finite sets is not valid in FSM. Theorem 3.2. There exists an infinite invariant set X (particularly the set A of atoms) such that every finitely supported surjective function from X onto itself is also one-to-one. Corollary 3.3. Let A be the infinite set of atoms in FSM. A finitely supported function f : A ! A is injective if and only if it is surjective. 4 Conclusion The goal of this paper is to enrich the results presented in [1] by men- tioning some consistency and inconsistency results regarding cardinal- ities and Dedekind-finiteness in Finitely Supported Mathematics. References [1] A. Alexandru, G. Ciobanu. Finitely Supported Mathematics: An Introduction. Springer, 2016. Andrei Alexandru, Gabriel Ciobanu1 1Romanian Academy, Institute of Computer Science, Iasi, Romania Email: [email protected] 6 Proceedings of the Conference on Mathematical Foundations of Informatics MFOI2017, November 9-11, 2017, Chisinau, Republic of Moldova Sequential Polarized Tissue P Systems with Vesicles of Multisets Artiom Alhazov, Rudolf Freund, Sergiu Ivanov, Sergey Verlan Abstract We consider sequential polarized tissue P systems working on vesicles of multisets with the very simple operations of inser- tion, deletion, and substitution of single objects. With the whole multiset being enclosed in a vesicle, sending it to a target cell can be indicated in those simple rules working on the multiset. Polarizations -1, 0, 1 are sufficient for obtaining computational completeness. 1 Introduction and Preliminaries For a comprehensive overview of different variants of (tissue) P systems and their expressive power we refer the reader to the handbook [4], and for a state of the art snapshot of the domain to the P systems web- site [6] as well as to the Bulletin series of the International Membrane Computing Society [5]. Very simple biologically motivated operations on strings are the so- called point mutations, i.e., insertion, deletion, and substitution.
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