Semantic Properties of T-consequence Relation in Logics of Quasiary Predicates

In the paper we investigate semantic properties of program-oriented algebras and logics defined for classes of quasiary predicates. Informally speaking, such predicates are partial predicates defined over partial states (partial assignments) of variables. Conventional n-ary predicates can be considered as a special case of quasiary predicates. We define first-order logics of quasiary non-deterministic predicates and investigate semantic properties of $T$-consequence relation for such logics. Specific properties of $T$-consequence relation for the class of deterministic predicates are also considered. Obtained results can be used to prove logic validity and completeness

Event-Based Proof of the Mutual Exclusion Property of Peterson’s Algorithm

Proving properties of distributed algorithms is still a highly challenging problem and various approaches that have been proposed to tackle it [1] can be roughly divided into state-based and event-based proofs. Informally speaking, state-based approaches define the behavior of a distributed algorithm as a set of sequences of memory states during its executions, while event-based approaches treat the behaviors by means of events which are produced by the executions of an algorithm. Of course, combined approaches are also possible.Analysis of the literature [1], [7], [12], [9], [13], [14], [15] shows that state-based approaches are more widely used than event-based approaches for proving properties of algorithms, and the difficulties in the event-based approach are often emphasized. We believe, however, that there is a certain naturalness and intuitive content in event-based proofs of correctness of distributed algorithms that makes this approach worthwhile. Besides, state-based proofs of correctness of distributed algorithms are usually applicable only to discrete-time models of distributed systems and cannot be easily adapted to the continuous time case which is important in the domain of cyber-physical systems. On the other hand, event-based proofs can be readily applied to continuous-time / hybrid models of distributed systems.In the paper [2] we presented a compositional approach to reasoning about behavior of distributed systems in terms of events. Compositionality here means (informally) that semantics and properties of a program is determined by semantics of processes and process communication mechanisms. We demonstrated the proposed approach on a proof of the mutual exclusion property of the Peterson’s algorithm [11]. We have also demonstrated an application of this approach for proving the mutual exclusion property in the setting of continuous-time models of cyber-physical systems in [8].Using Mizar [3], in this paper we give a formal proof of the mutual exclusion property of the Peterson’s algorithm in Mizar on the basis of the event-based approach proposed in [2]. Firstly, we define an event-based model of a shared-memory distributed system as a multi-sorted algebraic structure in which sorts are events, processes, locations (i.e. addresses in the shared memory), traces (of the system). The operations of this structure include a binary precedence relation ⩽ on the set of events which turns it into a linear preorder (events are considered simultaneous, if e1 ⩽ e2 and e2 ⩽ e1), special predicates which check if an event occurs in a given process or trace, predicates which check if an event causes the system to read from or write to a given memory location, and a special partial function “val of” on events which gives the value associated with a memory read or write event (i.e. a value which is written or is read in this event) [2]. Then we define several natural consistency requirements (axioms) for this structure which must hold in every distributed system, e.g. each event occurs in some process, etc. (details are given in [2]).After this we formulate and prove the main theorem about the mutual exclusion property of the Peterson’s algorithm in an arbitrary consistent algebraic structure of events. Informally, the main theorem states that if a system consists of two processes, and in some trace there occur two events e1 and e2 in different processes and each of these events is preceded by a series of three special events (in the same process) guaranteed by execution of the Peterson’s algorithm (setting the flag of the current process, writing the identifier of the opposite process to the “turn” shared variable, and reading zero from the flag of the opposite process or reading the identifier of the current process from the “turn” variable), and moreover, if neither process writes to the flag of the opposite process or writes its own identifier to the “turn” variable, then either the events e1 and e2 coincide, or they are not simultaneous (mutual exclusion property).Ievgen Ivanov - Taras Shevchenko National University, Kyiv, UkraineMykola Nikitchenko - Taras Shevchenko National University, Kyiv, UkraineUri Abraham - Ben-Gurion University, Beer-Sheva, IsraelUri Abraham. Models for Concurrency. Gordon and Breach, 1999.Uri Abraham, Ievgen Ivanov, and Mykola Nikitchenko. Proving behavioral properties of distributed algorithms using their compositional semantics. In Proceedings of the First International Seminar Specification and Verification of Hybrid Systems, October 10-12, 2011, Taras Shevchenko National University of Kyiv, pages 9–19, 2011.Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pąk, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8 17. [Crossref]Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.K. Chandy and J. Misra. Parallel Program Design: A Foundation. Addison Wesley, 1988.Ievgen Ivanov, Mykola Nikitchenko, and Uri Abraham. On a decidable formal theory for abstract continuous-time dynamical systems. In Vadim Ermolayev, Heinrich C. Mayr, Mykola Nikitchenko, Aleksander Spivakovsky, and Grygoriy Zholtkevych, editors, Information and Communication Technologies in Education, Research, and Industrial Applications, volume 469 of Communications in Computer and Information Science, pages 78–99. Springer International Publishing, 2014. ISBN 978-3-319-13205-1. doi:10.1007/978-3-319-13206-8 4. [Crossref]L. Lamport. On interprocess communication. Part I: Basic formalism; Part II: Algorithms. Distributed Computing, 1:77–101, 1986.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.G. Peterson. Myths about the mutual exclusion problem. Information Processing Letters, 12:1133–1145, 1981.V. Pratt. Modeling concurrency with partial orders. International Journal of Parallel Programming, 15:33–71, 1986.M. Raynal. A simple taxonomy for distributed mutual exclusion algorithms. ACM SIGOPS Operating Systems Review, 25:47–50, 1991.Tom Ridge. Peterson’s algorithm in Isabelle/HOL. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.99.3484, 2006.Tom Ridge. Operational reasoning for concurrent Caml programs and weak memory models. In Klaus Schneider and Jens Brandt, editors, Theorem Proving in Higher Order Logics, volume 4732 of Lecture Notes in Computer Science, pages 278–293. Springer Berlin Heidelberg, 2007. ISBN 978-3-540-74590-7. doi:10.1007/978-3-540-74591-4 21. [Crossref]Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski – Zorn lemma. Formalized Mathematics, 1(2):387–393, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990.Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85–89, 1990

On an Algorithmic Algebra over Simple-Named Complex-Valued Nominative Data

This paper continues formalization in the Mizar system [2, 1] of basic notions of the composition-nominative approach to program semantics [14] which was started in [8, 12, 10].The composition-nominative approach studies mathematical models of computer programs and data on various levels of abstraction and generality and provides tools for reasoning about their properties. In particular, data in computer systems are modeled as nominative data [15]. Besides formalization of semantics of programs, certain elements of the composition-nominative approach were applied to abstract systems in a mathematical systems theory [4, 6, 7, 5, 3].In the paper we give a formal definition of the notions of a binominative function over given sets of names and values (i.e. a partial function which maps simple-named complex-valued nominative data to such data) and a nominative predicate (a partial predicate on simple-named complex-valued nominative data). The sets of such binominative functions and nominative predicates form the carrier of the generalized Glushkov algorithmic algebra for simple-named complex-valued nominative data [15]. This algebra can be used to formalize algorithms which operate on various data structures (such as multidimensional arrays, lists, etc.) and reason about their properties.In particular, we formalize the operations of this algebra which require a specification of a data domain and which include the existential quantifier, the assignment composition, the composition of superposition into a predicate, the composition of superposition into a binominative function, the name checking predicate. The details on formalization of nominative data and the operations of the algorithmic algebra over them are described in [11, 13, 9].Ievgen Ivanov - Taras Shevchenko National University, Kyiv, UkraineArtur Korniłowicz - Institute of Informatics, University of Białystok, PolandMykola Nikitchenko - Taras Shevchenko National University, Kyiv, UkraineGrzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191–198, 2015. doi:10.1007/s10817-015-9345-1.Ievgen Ivanov. On the underapproximation of reach sets of abstract continuous-time systems. In Erika Ábrahám and Sergiy Bogomolov, editors, Proceedings 3rd International Workshop on Symbolic and Numerical Methods for Reachability Analysis, SNR@ETAPS 2017, Uppsala, Sweden, 22nd April 2017, volume 247 of EPTCS, pages 46–51, 2017. doi:10.4204/EPTCS.247.4.Ievgen Ivanov. On representations of abstract systems with partial inputs and outputs. In T. V. Gopal, Manindra Agrawal, Angsheng Li, and S. Barry Cooper, editors, Theory and Applications of Models of Computation – 11th Annual Conference, TAMC 2014, Chennai, India, April 11–13, 2014. Proceedings, volume 8402 of Lecture Notes in Computer Science, pages 104–123. Springer, 2014. ISBN 978-3-319-06088-0. doi:10.1007/978-3-319-06089-7_8.Ievgen Ivanov. On local characterization of global timed bisimulation for abstract continuous-time systems. In Ichiro Hasuo, editor, Coalgebraic Methods in Computer Science – 13th IFIP WG 1.3 International Workshop, CMCS 2016, Colocated with ETAPS 2016, Eindhoven, The Netherlands, April 2–3, 2016, Revised Selected Papers, volume 9608 of Lecture Notes in Computer Science, pages 216–234. Springer, 2016. ISBN 978-3-319-40369-4. doi:10.1007/978-3-319-40370-0_13.Ievgen Ivanov, Mykola Nikitchenko, and Uri Abraham. On a decidable formal theory for abstract continuous-time dynamical systems. In Vadim Ermolayev, Heinrich C. Mayr, Mykola Nikitchenko, Aleksander Spivakovsky, and Grygoriy Zholtkevych, editors, Information and Communication Technologies in Education, Research, and Industrial Applications: 10th International Conference, ICTERI 2014, Kherson, Ukraine, June 9–12, 2014, Revised Selected Papers, pages 78–99. Springer International Publishing, 2014. ISBN 978-3-319-13206-8. doi:10.1007/978-3-319-13206-8_4.Ievgen Ivanov, Mykola Nikitchenko, and Uri Abraham. Event-based proof of the mutual exclusion property of Peterson’s algorithm. Formalized Mathematics, 23(4):325–331, 2015. doi:10.1515/forma-2015-0026.Ievgen Ivanov, Mykola Nikitchenko, Andrii Kryvolap, and Artur Korniłowicz. Simple-named complex-valued nominative data – definition and basic operations. Formalized Mathematics, 25(3):205–216, 2017. doi:10.1515/forma-2017-0020.Ievgen Ivanov, Artur Korniłowicz, and Mykola Nikitchenko. Implementation of the composition-nominative approach to program formalization in Mizar. The Computer Science Journal of Moldova, 26(1):59–76, 2018.Ievgen Ivanov, Artur Korniłowicz, and Mykola Nikitchenko. On algebras of algorithms and specifications over uninterpreted data. Formalized Mathematics, 26(2):141–147, 2018. doi:10.2478/forma-2018-0011.Artur Kornilowicz, Andrii Kryvolap, Mykola Nikitchenko, and Ievgen Ivanov. Formalization of the algebra of nominative data in Mizar. In Maria Ganzha, Leszek A. Maciaszek, and Marcin Paprzycki, editors, Proceedings of the 2017 Federated Conference on Computer Science and Information Systems, FedCSIS 2017, Prague, Czech Republic, September 3–6, 2017., pages 237–244, 2017. ISBN 978-83-946253-7-5. doi:10.15439/2017F301.Artur Korniłowicz, Ievgen Ivanov, and Mykola Nikitchenko. Kleene algebra of partial predicates. Formalized Mathematics, 26(1):11–20, 2018. doi:10.2478/forma-2018-0002.Artur Korniłowicz, Andrii Kryvolap, Mykola Nikitchenko, and Ievgen Ivanov. Formalization of the nominative algorithmic algebra in Mizar. In Jerzy Świątek, Leszek Borzemski, and Zofia Wilimowska, editors, Information Systems Architecture and Technology: Proceedings of 38th International Conference on Information Systems Architecture and Technology – ISAT 2017: Part II, pages 176–186. Springer International Publishing, 2018. ISBN 978-3-319-67229-8. doi:10.1007/978-3-319-67229-8_16.Nikolaj S. Nikitchenko. A composition nominative approach to program semantics. Technical Report IT-TR 1998-020, Department of Information Technology, Technical University of Denmark, 1998.Volodymyr G. Skobelev, Mykola Nikitchenko, and Ievgen Ivanov. On algebraic properties of nominative data and functions. In Vadim Ermolayev, Heinrich C. Mayr, Mykola Nikitchenko, Aleksander Spivakovsky, and Grygoriy Zholtkevych, editors, Information and Communication Technologies in Education, Research, and Industrial Applications – 10th International Conference, ICTERI 2014, Kherson, Ukraine, June 9–12, 2014, Revised Selected Papers, volume 469 of Communications in Computer and Information Science, pages 117–138. Springer, 2014. ISBN 978-3-319-13205-1. doi:10.1007/978-3-319-13206-8_6.26214915

Kleene Algebra of Partial Predicates

We show that the set of all partial predicates over a set D together with the disjunction, conjunction, and negation operations, defined in accordance with the truth tables of S.C. Kleene’s strong logic of indeterminacy [17], forms a Kleene algebra. A Kleene algebra is a De Morgan algebra [3] (also called quasi-Boolean algebra) which satisfies the condition x ∧¬:x ⩽ y ∨¬ :y (sometimes called the normality axiom). We use the formalization of De Morgan algebras from [8]. The term “Kleene algebra” was introduced by A. Monteiro and D. Brignole in [3]. A similar notion of a “normal i-lattice” had been previously studied by J.A. Kalman [16]. More details about the origin of this notion and its relation to other notions can be found in [24, 4, 1, 2]. It should be noted that there is a different widely known class of algebras, also called Kleene algebras [22, 6], which generalize the algebra of regular expressions, however, the term “Kleene algebra” used in this paper does not refer to them. Algebras of partial predicates naturally arise in computability theory in the study on partial recursive predicates. They were studied in connection with non-classical logics [17, 5, 18, 32, 29, 30]. A partial predicate also corresponds to the notion of a partial set [26] on a given domain, which represents a (partial) property which for any given element of this domain may hold, not hold, or neither hold nor not hold. The field of all partial sets on a given domain is an algebra with generalized operations of union, intersection, complement, and three constants (0, 1, n which is the fixed point of complement) which can be generalized to an equational class of algebras called DMF-algebras (De Morgan algebras with a single fixed point of involution) [25]. In [27] partial sets and DMF-algebras were considered as a basis for unification of set-theoretic and linguistic approaches to probability. Partial predicates over classes of mathematical models of data were used for formalizing semantics of computer programs in the composition-nominative approach to program formalization [31, 28, 33, 15], for formalizing extensions of the Floyd-Hoare logic [7, 9] which allow reasoning about properties of programs in the case of partial pre- and postconditions [23, 20, 19, 21], for formalizing dynamical models with partial behaviors in the context of the mathematical systems theory [11, 13, 14, 12, 10].Korniłowicz Artur - Institute of Informatics, University of Białystok, PolandIvanov Ievgen - Taras Shevchenko National University, Kyiv, UkraineNikitchenko Mykola - Taras Shevchenko National University, Kyiv, UkraineRaymond Balbes and Philip Dwinger. Distributive Lattices. University of Missouri Press, 1975.T.S. Blyth and J. Varlet. Ockham Algebras. Oxford science publications. Oxford University Press, 1994.Diana Brignole and Antonio Monteiro. Caractérisation des algèbres de Nelson par des egalités. Instituto de Matemática, Universidad Nacional del Sur, Argentina, 1964.Roberto Cignoli. Injective de Morgan and Kleene algebras. Proceedings of the American Mathematical Society, 47(2):269–278, 1975.J.P. Cleave. A Study of Logics. Oxford logic guides. Clarendon Press, 1991. ISBN 9780198532118.J.H. Conway. Regular algebra and finite machines. Chapman and Hall mathematics series. Chapman and Hall, 1971.R.W. Floyd. Assigning meanings to programs. Mathematical aspects of computer science, 19(19–32), 1967.Adam Grabowski. Robbins algebras vs. Boolean algebras. Formalized Mathematics, 9(4): 681–690, 2001.C.A.R. Hoare. An axiomatic basis for computer programming. Commun. ACM, 12(10): 576–580, 1969.Ievgen Ivanov. On the underapproximation of reach sets of abstract continuous-time systems. In Erika Ábrahám and Sergiy Bogomolov, editors, Proceedings 3rd International Workshop on Symbolic and Numerical Methods for Reachability Analysis, SNR@ETAPS 2017, Uppsala, Sweden, 22nd April 2017, volume 247 of EPTCS, pages 46–51, 2017. doi:10.4204/EPTCS.247.4.Ievgen Ivanov. On representations of abstract systems with partial inputs and outputs. In T. V. Gopal, Manindra Agrawal, Angsheng Li, and S. Barry Cooper, editors, Theory and Applications of Models of Computation – 11th Annual Conference, TAMC 2014, Chennai, India, April 11–13, 2014. Proceedings, volume 8402 of Lecture Notes in Computer Science, pages 104–123. Springer, 2014. ISBN 978-3-319-06088-0. doi:10.1007/978-3-319-06089-7_8.Ievgen Ivanov. On local characterization of global timed bisimulation for abstract continuous-time systems. In Ichiro Hasuo, editor, Coalgebraic Methods in Computer Science – 13th IFIP WG 1.3 International Workshop, CMCS 2016, Colocated with ETAPS 2016, Eindhoven, The Netherlands, April 2–3, 2016, Revised Selected Papers, volume 9608 of Lecture Notes in Computer Science, pages 216–234. Springer, 2016. ISBN 978-3-319-40369-4. doi:10.1007/978-3-319-40370-0_13.Ievgen Ivanov, Mykola Nikitchenko, and Uri Abraham. On a Decidable Formal Theory for Abstract Continuous-Time Dynamical Systems, pages 78–99. Springer International Publishing, 2014. ISBN 978-3-319-13206-8. doi:10.1007/978-3-319-13206-8_4.Ievgen Ivanov, Mykola Nikitchenko, and Uri Abraham. Event-based proof of the mutual exclusion property of Peterson’s algorithm. Formalized Mathematics, 23(4):325–331, 2015. doi:10.1515/forma-2015-0026.Ievgen Ivanov, Mykola Nikitchenko, and Volodymyr G. Skobelev. Proving properties of programs on hierarchical nominative data. The Computer Science Journal of Moldova, 24(3):371–398, 2016.J. A. Kalman. Lattices with involution. Transactions of the American Mathematical Society, 87(2):485–485, February 1958. doi:10.1090/s0002-9947-1958-0095135-x.S.C. Kleene. Introduction to Metamathematics. North-Holland Publishing Co., Amsterdam, and P. Noordhoff, Groningen, 1952.S. Körner. Experience and Theory: An Essay in the Philosophy of Science. International library of philosophy and scientific method. Routledge & Kegan Paul, 1966.Artur Kornilowicz, Andrii Kryvolap, Mykola Nikitchenko, and Ievgen Ivanov. Formalization of the algebra of nominative data in Mizar. In Maria Ganzha, Leszek A. Maciaszek, and Marcin Paprzycki, editors, Proceedings of the 2017 Federated Conference on Computer Science and Information Systems, FedCSIS 2017, Prague, Czech Republic, September 3–6, 2017., pages 237–244, 2017. ISBN 978-83-946253-7-5. doi:10.15439/2017F301.Artur Korniłowicz, Andrii Kryvolap, Mykola Nikitchenko, and Ievgen Ivanov. An approach to formalization of an extension of Floyd-Hoare logic. In Vadim Ermolayev, Nick Bassiliades, Hans-Georg Fill, Vitaliy Yakovyna, Heinrich C. Mayr, Vyacheslav Kharchenko, Vladimir Peschanenko, Mariya Shyshkina, Mykola Nikitchenko, and Aleksander Spivakovsky, editors, Proceedings of the 13th International Conference on ICT in Education, Research and Industrial Applications. Integration, Harmonization and Knowledge Transfer, Kyiv, Ukraine, May 15–18, 2017, volume 1844 of CEUR Workshop Proceedings, pages 504–523. CEUR-WS.org, 2017.Artur Korniłowicz, Andrii Kryvolap, Mykola Nikitchenko, and Ievgen Ivanov. Formalization of the Nominative Algorithmic Algebra in Mizar, pages 176–186. Springer International Publishing, 2018. ISBN 978-3-319-67229-8. doi:10.1007/978-3-319-67229-8_16.Dexter Kozen. On Kleene algebras and closed semirings, pages 26–47. Springer Berlin Heidelberg, 1990. doi:10.1007/BFb0029594.Andrii Kryvolap, Mykola Nikitchenko, and Wolfgang Schreiner. Extending Floyd-Hoare Logic for Partial Pre- and Postconditions, pages 355–378. Springer International Publishing, 2013. ISBN 978-3-319-03998-5. doi:10.1007/978-3-319-03998-5_18.Antonio Monteiro and Luiz Monteiro. Axiomes indépendants pour les algèbres de Nelson, de Łukasiewicz trivalentes, de de Morgan et de Kleene. Notas de lógica matemática, (40): 1–11, 1996.Maurizio Negri. Three valued semantics and DMF-algebras. Boll. Un. Mat. Ital. B (7), 10(3):733–760, 1996.Maurizio Negri. DMF-algebras: representation and topological characterization. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1(2):369–390, 1998.Maurizio Negri. Partial probability and Kleene logic, 2013.M.S. Nikitchenko and S.S. Shkilniak. Mathematical logic and theory of algorithms. Publishing house of Taras Shevchenko National University of Kyiv, Ukraine (in Ukrainian), 2008.M.S. Nikitchenko and S.S. Shkilniak. Applied logic. Publishing house of Taras Shevchenko National University of Kyiv, Ukraine (in Ukrainian), 2013.Mykola Nikitchenko and Stepan Shkilniak. Algebras and logics of partial quasiary predicates. Algebra and Discrete Mathematics, 23(2):263–278, 2017.Nikolaj S. Nikitchenko. A composition nominative approach to program semantics. Technical Report IT-TR 1998-020, Department of Information Technology, Technical University of Denmark, 1998.Helena Rasiowa. An Algebraic Approach to Non-Classical Logics. North Holland, 1974.Volodymyr G. Skobelev, Mykola Nikitchenko, and Ievgen Ivanov. On algebraic properties of nominative data and functions. In Vadim Ermolayev, Heinrich C. Mayr, Mykola Nikitchenko, Aleksander Spivakovsky, and Grygoriy Zholtkevych, editors, Information and Communication Technologies in Education, Research, and Industrial Applications – 10th International Conference, ICTERI 2014, Kherson, Ukraine, June 9–12, 2014, Revised Selected Papers, volume 469 of Communications in Computer and Information Science, pages 117–138. Springer, 2014. ISBN 978-3-319-13205-1. doi:10.1007/978-3-319-13206-8_6.261112

Set-theoretic Analysis of Nominative Data

In the paper we investigate the notion of nominative data that can be considered as a general mathematical model of data used in computing systems. The main attention is paid to flat nominative data called nominative sets. The structure of the partially-ordered set of nominative sets is investigated in terms of set theory, lattice theory, and algebraic systems theory. To achieve this aim the correct transferring of basic set-theoretic operations to nominative sets is proposed. We investigate a lower semilattice of nominative sets in terms of lower and upper cones, closed and maximal closed intervals of nominative sets. The obtained results can be used in formal software development