This survey article illustrates many important current trends and perspectives for the field and their applications, of interest to researchers in modern algebra, mathematical logic and discrete mathematics. It covers a number of directions, including completeness theorem and compactness theorem for hyperidentities; the characterizations of the Boolean algebra of n -ary Boolean functions and the bounded distributive lattice of n -ary monotone Boolean functions; the functional representations of finitely-generated free algebras of various varieties of lattices via generalized Boolean functions, etc.
KeywordsBoolean AlgebraBoolean FunctionDistributive LatticeMonotone Boolean FunctionDe Morgan AlgebraDe Morgan FunctionBoole-De Morgan AlgebraQuasi-De Morgan FunctionHyperidentityCoidentity
Boole, G. (1853) The Laws of Thought. Dover.
Dedekind, R. (1897) Über Zerlegungen von Zahlen Durch Ihre Grössten Gemeinsamen Theiler. In: Dedekind, R., ED., Fest-Schrift der Herzoglichen Technischen Hochschule Carolo-Wilhelmina , Vieweg+Teubner Verlag, 1-40. https://doi.org/10.1007/978-3-663-07224-9_1
Church, A. (1956) Introduction to Mathematical Logic. Vol. 1. Princeton University Press.
Mal’tsev, A.I. (1963) On the General Theory of Algebraic Systems. AMS American Mathematical Society Translations , 27, 125-142.
Mal’tsev, A.I. (1963) Some Questions of the Theory of Classes of Models. Proceedings of the IVth All-Union Mathematical Congress , 1, 169-198. (In Russian)
Mal’tsev, A.I. and Mal’tsev, I.A. (2012) Iterative Post algebras. Nauka. (In Russian)
Grätzer, G. (1979) Universal Algebra. 2nd Edition, Springer-Verlag.
Chang, C.C. and Keisler, H.J. (1973) Model Theory. Studies in Logic and the Foundations of Mathematics . Vol. 73, Nort-Holland Publishing Co., American Elsevier Publishing Co. Inc.
Burris, S. and Sankappanavar, H.P. (1981) A Course in Universal Algebra. Springer-Verlag.
Kogalovskii, S.R. (1961) On Multiplicative Semigroups of Rings. Proceedings of the USSR Academy of Sciences , 140, 1005-1007. (In Russian)
Sabbagh, G. (1969) How Not to Characterize the Multiplicative Groups of Fields. Journal of the London Mathematical Society , 2, 369-370. https://doi.org/10.1112/jlms/s2-1.1.369
Hall Jr., M. (1959) The Theory of Groups. Chelsea.
Grätzer, G. (1964) A Theorem on Doubly Transitive Permutation Groups with Application to Universal Algebras. Fundamenta Mathematicae , 53, 25-41. https://doi.org/10.4064/fm-53-1-25-41
Ganter, B. and Werner, H. (1975) Equational Classes of Steiner Systems. Algebra Universalis , 5, 125-140. https://doi.org/10.1007/bf02485242
Belousov, V.D. (1966) Gratzer Algebra and S-Systems of Quasigroups. Mat. Issledovaniya Akad . Nauk Moldavii , Kishinev , 1, 55-81. (In Russian)
Movsisyan, Y.M. (2017) Hyperidentities and Related Concepts I. Armenian Journal of Mathematics , 2, 146-222.
Pontryagin, L.S. (1973) Continuous Groups. Nauka. (In Russian)
Movsisyan, Y.M. (1990) Hyperidentities and Hypervarieties in Algebras. Yerevan State University Press.
Fuchs, L. (1973) Infinite Abelian Groups, Volume II. Academic Press.
Kargapolov, M.I. and Merzlyakov, Y.I. (1977) Foundation of Group Theory. Nauka.
Schauffler, R. (1956) Uber die Bildung von Codewörtern. Archiv der elektrischen Übertragung , 10, 303-314.
Schauffler, R. (1957) Die Assoziativität im Ganzen, besonders bei Quasigruppen. Mathematische Zeitschrift , 67, 428-435. https://doi.org/10.1007/bf01258874
Sade, A. (1960) Théorie des systèmes demosiens de groupoï des. Pacific Journal of Mathematics , 10, 625-660. https://doi.org/10.2140/pjm.1960.10.625
Sade, A. (1961) Groupoides en relation associative et semigroupes mutuellements associatifs. Annales de la Société scientifique de Bruxelles , 1, 52-57.
Belousov, V.D. (1965) Systems of Quasigroups with Generalized Identities. Russian Mathematical Surveys , 20, 75-143. https://doi.org/10.1070/rm1965v020n01abeh004140
Denes, J., and Keedwell, A.D. (1974) Lattin Squares and Their Applications. Akademiai Kiado. (Also Academic Press and English Universities Press).
Movsisyan, Y.M. (1974) Biprimitive Classes of Algebras of Second Degree. M a tematicheskie Issledovaniya , 9, 70-84. (in Russian)
Movsisyan, Y.M. (1983) Coidentities in algebras. Doklady Akademii Nauk Armyanskoi SSR , 77, 51-54.
Movsisyan, Y.M. and Davidov, S.S. (2018) Algebras That Are Nearly Quasigroups. LENAND. (In Russian)
Belousov, V.D. and Movsisyan, Y.M. (1974) On the Rank of Generalized Identities. Izv . Acad. Nauk . Armenii , Ser. Mat. , 9, 135-142.
Čupona, G. (1959-1961) Za finitarnite operazii. Godishen zbornik univerzitetot Skopje , Prirod - no- Matematichki Fakultet , 1, 7-49.
Taylor, W. (1981) Hyperidentities and hypervarieties. Aequationes Mathematicae , 23, 30-49. https://doi.org/10.1007/bf02188010
Bergman, G.M. (1981) Hyperidentities of Groups and Semigroups. Aequationes Mathematicae , 23, 50-65. https://doi.org/10.1007/bf02188011
Bergman, G.M. (2015) An Invitation on General Algebra and Universal Constructions. 2nd Edition, Springer.
Padmanabhan, R. and Penner, P. (1982) Bases of Hyperidentities of Lattices and Semilattices. C omptes Rendus Math. Acad. Sci. Canada , 4, 9-14.
Movsisyan, Y.M. (1986) Introduction to the Theory of Algebras with Hyperidentities. Yerevan State University Press. (In Russian)
Movsisyan, Y.M. (2001) Hyperidentities and Hypervarieties. Scientiae Mathematicae Japonicae , 54, 595-640.
Movsisyan, Y. (2022) Hyperidentities and Related Concepts, II. Armenian Journal of Mathematics , 10, 1-85. https://doi.org/10.52737/18291163-2018.10.4-1-85
Movsisyan, Y.M. (2018) On Functional Equations and Distributive Second-Order Formulae with Specialized Quantifiers. Algebra and Discrete Mathematics , 25, 269-285.
Smith, J.D.H. (2010) On Groups of Hypersubstitutions. Algebra universalis , 64, 39-48. https://doi.org/10.1007/s00012-010-0087-y
Padmanabhan, R. and Penner, P. (1992) Binary Hyperidentities of Lattices. Aequationes Mathematicae , 44, 154-167. https://doi.org/10.1007/bf01830976
Słomiński, J. (1993) On the Satisfiabilities and Varieties for Abstract Algebras Induced by the Cones and Functor Dynamics. Demonstratio Mathematica , 26, 11-22. https://doi.org/10.1515/dema-1993-0103
Movsisyan, Y.M. (1993) On a Theorem of Schauffler. Mathematical Notes , 53, 172-179. https://doi.org/10.1007/bf01208322
Movsisyan, Y.M., Romanowska, A.B. and Smith, J.D.H. (2004) Hyperidentities and Ginsberg-Fitting Type Theorems. Eighteenth Midwest Conference on Combinatorics , Cryptography , and Computing , Rochester, 28-30 October 2004, 15.
Movsisyan, Y.M., Romanowska, A.B. and Smith, J.D.H. (2006) Superproducts, Hyperidentities, and Algebraic Structures of Logic Programming. The Journal of Combinatorial Mathematics and Combinatorial Computing , 58, 101-111.
Krapež, A. (1980) Generalized Associativity on Groupoids. Publications de l ’ Institut Mathématique ( Beograd ), 28, 105-112.
Schweigert, D. (1993) Hyperidentities. In: Rosenberg, I.G. and Sabidussi, G., Eds., Algebras and Orders , Springer Netherlands, 405-506. https://doi.org/10.1007/978-94-017-0697-1_10
Mal’tsev, I.A. and Schveigert, D. (1989) Hyperidentitäten von AZ-algebren. Fachbereich Mathematik, Universität Kaiserslautern.
Mando, M. (1990) Regular-Identities of Length Four in Algebras. Ph.D. Thesis,.Yerevan State University.
Paseman, G. (2014) On Two Problems from “Hyperidentities and Clones”. https://arxiv.org/abc/1408.2784
Denecke, K. and Wismath, S.L. (2000) Hyperidentities and Clones. Gordon and Breach Science Publishers.
Graczynska, E. (2014) Algebra of M-Solid Quasivarieties. Siatras International Bookshop.
Graczynska, E. (2007) On the Problem on M-Hyperquasivarieties. https://arxiv.org/abc/math/0609600
Ušan, J. (1975) Globally Associative Systems of N-Ary Quasigroups. Publications de l ’ Institut Mathematique ( Beograd ), 19, 155-165.
Usan, J. (2001) Description of Super Associative Algebras with N-Quasigroup Operations. Mathematica Moravica , 2001, 129-157. https://doi.org/10.5937/matmor0105129u
Usan, J. (2006) (n, m)-Groups in the Light of the Neutral Operations Survey Article. Mathematica Moravica , 2006, 107-147. https://doi.org/10.5937/matmor0610147u
Ušan, J. and Žižović, M. (1975) N-Ary Analogy of Schauffler Theorem. Publications de l ’ Institut Mathematique ( Beograd ), 19, 167-172.
Žarkov, D. (1977) A Remark on Globally Associative Systems of N-Quasigroups over . Publications de l ’ Institut Mathematique ( Beograd ), 21, 207-211.
Belyavskaya, G.B. (1974) S-Systems of Quasigroups. Mat. Issledovaniya Acad. Nauk Moldavii , Kishinev , 9, 10-18.
Movsisyan, Y. (2022) Hyperidentities: Boolean and De Morgan Structures. World Scientific. https://doi.org/10.1142/12796
Koppitz, J. and Denecke, K. (2006) M-Solid Varieties of Algebras. Springer.
DeMorgan, A. (1847) Formal Logic. Taylor and Walton.
Denecke, K. (1998) Hyperequational Theory. Mathematica Japonica , 47, 333-356.
Denecke, K. and Koppitz, J. (1994) Hyperassociative Varieties of Semigroups. Semigroup Forum , 49, 41-48. https://doi.org/10.1007/bf02573469
Denecke, K., Lau, D., Pöschel, R. and Schweigert, D. (1991) Hyperidentities, Hyperequational Classes, and Clone Congruences. Contribution to General Algebr , Verlag Hölder-Pichler-Tempsky, 97-118.
Graczyńska, E. and Schweigert, D. (1990) Hypervarieties of a Given Type. Algebra Universalis , 27, 305-318. https://doi.org/10.1007/bf01190711
Skornyakov, L.A. (1991) General Algebra, Vol. 2. Nauka. (In Russian)
Belousov, V.D. (1961) Globally Associative Systems of Quasigroups. Matematicheskii Sbornik , 55, 221-236.
Kurosh, A.G. (1963) Lectures on General Algebra. Chelsea.
Plotkin, B.I. (1966) Automorphisms Groups of Algebraic Systems. Nauka. (In Russian)
Adian, S.I. (1979) The Burnside Problem and Identities in Groups. Springer.
Adian, S.I. (2011) The Burnside Problem and Related Topics. Russian Mathematical Surveys , 65, 805-855. https://doi.org/10.1070/rm2010v065n05abeh004702
Zel’manov, E.I. (1990) The Solution of the Restricted Burnside Problem for Groups of Odd Exponent. Math. USSR Izv , 54, 42-59.
Zel’manov, E.I. (1991) The Solution of the Restricted Burnside Problem for 2-Groups. Matematicheskii Sbornik , 182, 568-592.
Zelmanov, E.I. (1993) On Additional Laws in the Burnside Problem on Periodic Groups. International Journal of Algebra and Computation , 3, 583-600. https://doi.org/10.1142/s0218196793000330
Zelmanov, E. (2007) Some Open Problems in the Theory of Infinite Dimensional Algebras. Journal of the Korean Mathematical Society , 44, 1185-1195. https://doi.org/10.4134/jkms.2007.44.5.1185
Movsisyan, Y.M. (2016) On Burnside Varieties. VII International Joint Conference of the Georgian Mathematical Union & Georgian Mechanical Union ( Dedicated to 125 th Birthday Anniversary of Academician N. Muskhelishvili ), Batumi, 5-9 September 2016, 172.
Burnside, W. (1902) On an Unsettled Question in the Theory of Discountinuous Groups. The Quarterly Journal of Pure and Applied Mathematics , 33, 230-238.
Sanov, I.N. (1940) Solution of Burnside’s Problem for Exponent 4. Scientific Notices of Leningrad State University , 10, 166-170. (In Russian)
Novikov, P.S. and Adian, S.I. (1968) Infinite Periodic Groups. Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya , 32, 212-244, 251-524, 709-731.
Lysenok, I.G. (1992) The Infinitude of Burnside Groups of Period 2 k for . Russian Mathematical Surveys , 47, 229-230. https://doi.org/10.1070/rm1992v047n02abeh000888
Kurosh, A.G. (1974) General Algebra. Nauka. (In Russian)
Romanowska, A.B. and Smith, J.D.H. (1985) Modal Theory. Helderman-Verlag.
Romanowska, A.B. and Smith, J.D.H. (2002) Modes. World Scientific.
Knoebel, A. and Romanowska, A. (1991) Distributive Multisemilattices. Instytut Matematyczny Polskiej Akademi Nauk.
Aguiar, M. and Livernet, M. (2007) The Associative Operad and the Weak Order on the Symmetric Groups. Journal of Homotopy and Related Structures , 2, 57-84.
Loday, J. and Ronco, M.O. (2002) Order Structure on the Algebra of Permutations and of Planar Binary Trees. Journal of Algebraic Combinatorics , 15, 253-270. https://doi.org/10.1023/a:1015064508594
Richter, B. (1997) Dialgebren, Doppelalgebren und ihre Homologie, Diplomarbeit, Universitat Bonn. http://www.math.uni-hamburg.de/home/richter/publications.html
Zhuchok, Y.V. (2016) The Endomorphism Monoid of a Free Trioid of Rank 1. Algebra universalis , 76, 355-366. https://doi.org/10.1007/s00012-016-0392-1
Zhuchok, A.V. and Demko, M. (2016) Free n-Dinilpotent Doppelsemigroups. Algebra and Discrete Mathematics , 22, 304-316.
Zhuchok, A.V. and Knauer, K. (2018) Abelian doppelsemigroups. Algebra and Discrete Mathematics , 26, 290-304.
Zhuchok, A.V. (2018) Structure of Free Strong Doppelsemigroups. Communications in Algebra , 46, 3262-3279. https://doi.org/10.1080/00927872.2017.1407422
Zhuchok, A.V. (2018) Relatively Free Doppelsemigroups. Potsdam University Press.
Zhuchok, A.V., Zhuchok, Y.V. and Koppitz, J. (2019) Free Rectangular Doppelsemigroups. Journal of Algebra and Its Applications , 19, Article ID: 2050205. https://doi.org/10.1142/s0219498820502059
Barnes, W. (1966) On the γ -Rings of Nobusawa. Pacific Journal of Mathematics , 18, 411-422. https://doi.org/10.2140/pjm.1966.18.411
Hedayati, H. and Shum, K.P. (2011) An Introduction to γ -Semirings. International Journal of Algebra , 5, 709-726.
Luh, J. (1969) On the Theory of Simple γ -Rings. Michigan Mathematical Journal , 16, 65-75. https://doi.org/10.1307/mmj/1029000167
Nobusawa, N. (1964) On a Generalization of the Ring Theory. Osaka Journal of Mathematics , 1, 81-89.
Muralikrishna Rau, M. (1995) Г-Semirings-I. Southeast Asian Bull. Math. , 19, 49-54.
Sardar, S.K., Gupta, S. and Shum, K.P. (2013) γ -Semigroups with Unites and Morita Equivalence for Monoids. European Journal of Pure and Applied Mathematics , 6, 1-10.
Sardar, S.K., Saha, B.C. and Shum, K.P. (2010) h-Prime and h-Semiprime Ideals in Semirings and γ -Semirings. International Journal of Algebra , 4, 209-220.
Sen, M.K. and Saha, N.K. (1986) On γ -Semigroup—I. Bulletin of the Calcutta Mathematical Society , 78, 180-186.
Seth, A. (1990) γ ‐Group Congruences on Regular γ ‐Semigroups. International Journal of Mathematics and Mathematical Sciences , 15, 103-106. https://doi.org/10.1155/s0161171292000115
Muralikrishna Rau, M. (1997) Г-Semirings—II. Southeast Asian Bull. Math. , 21, 281-287.
Kehayopulu, N. (2011) On Regular Duo Po- γ -Semigroups. Mathematica Slovaca , 61, 871-884. https://doi.org/10.2478/s12175-011-0054-x
Denecke, K. and Pöschel, R. (1988) A Characterization of Sheffer Functions by Hyperidentities. Semigroup Forum , 37, 351-362. https://doi.org/10.1007/bf02573146
Denecke, K. and Pöschel, R. (1988) The Characterization of Primal Algebras by Hyperidentities. In: Dorninger, D., Ed., Contributions to General Algebra 6, Hölder-Pichler-Tempsky, 67-87.
Melkonian, V. (2011) Circuit Integration through Lattice Hyperterms. Discrete Mathematics , Algorithms and Applications , 03, 101-119. https://doi.org/10.1142/s179383091100105x
Anosov, A.D. (2007) On Homomorphisms of Many-Sorted Algebraic Systems in Connection with Cryptographic Applications. Discrete Mathematics and Applications , 17, 331-347. https://doi.org/10.1515/dma.2007.028
Aczél, J. (1971) Proof of a Theorem on Distributive Type Hyperidentities. Algebra Universalis , 1, 1-6. https://doi.org/10.1007/bf02944948
Movsisyan, Y.M. (1990) The Multiplicative GROUP of a field and Hyperidentities. Mathematics of the USSR- Izvestiya , 35, 377-391. https://doi.org/10.1070/im1990v035n02abeh000708
Movsisyan, Y.M. (1993) Hyperidentities of Boolean Algebras. Russian Academy of Sciences. Izvestiya Mathematics , 40, 607-622. https://doi.org/10.1070/im1993v040n03abeh002179
Movsisyan, Y.M. (1996) Algebras with Hyperidentities of the Variety of Boolean Algebras. Izvestiya : Mathematics , 60, 1219-1260. https://doi.org/10.1070/im1996v060n06abeh000098
Movsisyan, Y.M. (1996) Superidentities in the Variety of Lattices. Mathematical Notes , 59, 686-688. https://doi.org/10.1007/bf02307222
Penner, P. (1984) Hyperidentities of Semilattices. Houston Journal of Mathematics , 10, 81-108.
Movsisyan, Y.M. (2011) Bilattices and Hyperidentities. Proceedings of the Steklov Institute of Mathematics , 274, 174-192. https://doi.org/10.1134/s0081543811060113
Movsisyan, Y.M. (2008) Interlaced, Modular, Distributive and Boolean Bilattices. Armenian Journal of Mathematics , 1, 7-13.
Movsisyan, Y.M. (2022) Boole-De Morgan Bilattices. Journal of Multiple - Valued Logic and Soft Computing , 38, 137-152.
Birkhoff, G. (1944) Subdirect Unions in Universal Algebra. Bulletin of the American Mathematical Society , 50, 764-768. https://doi.org/10.1090/s0002-9904-1944-08235-9
Birkhoff, G. (1935) On the Structure of Abstract Algebras. Mathematical Proceedings of the Cambridge Philosophical Society , 31, 433-454. https://doi.org/10.1017/s0305004100013463
Crama, Y. and Hammer, P.L. (2011) Boolean Functions: Theory, Algorithms, and Applications. Cambridge University Press. https://doi.org/10.1017/cbo9780511852008
Birkhoff, G. (1979) Lattice Theory. 3rd Edition, American Mathematical Society.
Balbes, R. and Dwinger, P. (1974) Distributive Lattices. University of Missouri Press.
Bialynicki-Birula, A. and Rasiowa, H. (1957) On the Representation of Quasi-Boolean Algebras. Bulletin L ’ Académie Polonaise des Science , Série des Sciences Mathématiques , Astronomiques et Physiques , 5, 259-261.
Brzozowski, J.A. (2000) De Morgan Bisemilattices. The 30 th IEEE International Sym- pozium on Multiple - Valued Logic , Portland, May 2000, 23-25.
Brzozowski, J.A. (2001) A Characterization of de Morgan Algebras. International Journal of Algebra and Computation , 11, 525-527. https://doi.org/10.1142/s0218196701000681
Brzozowski, J.A. and Seger, C.J.H. (1995) Asynchronous Circuits. Springer-Verlag.
Ésik, Z. (1998) A Cayley Theorem for Ternary Algebras. International Journal of Algebra and Computation , 8, 311-316. https://doi.org/10.1142/s0218196798000156
Kalman, J.A. (1958) Lattices with Involution. Transactions of the American Mathematical Society , 87, 485-491. https://doi.org/10.1090/s0002-9947-1958-0095135-x
Markov, A.A. (1950) Constructive Logic. Uspekhi Matematicheskikh Nauk , 5, 187-188. (In Russian)
Moisil, G.C. (1935) Recherches sur l’algebre de la logique. Annales scientifiques de l ’ uni - versite de Jassy , 22, 1-117.
Rasiowa, H. (1974) An Algebraic Approach to Non-Classical Logics. Nort-Holland Publisher.
Sankappanavar, H.P. (1980) A Characterization of Principal Congruences of De Morgan Algebras and Its Applications Studies in Logic and the Foundations of Mathematics , 99, 341-349. https://doi.org/10.1016/s0049-237x(09)70493-8
Sankappanavar, H.P. (2018) Implication Zroupoids: An Abstraction from De Morgan Algebras. Emil Artin International Conference dedicated to th e 120 th Anniversary of Emil Artin , Yerevan, 27 May-2 June 2018, 119-120.
Movsisyan, Y.M. (2005) Boolean Bisemigroups, Bigroups and Local Bigroups. Computer Science and Information Technologies , Proceedings of the Conference , Yerevan, 19-20 September 2005, 97-104.
Movsisyan, Y.M. (2005) On the Representations of De Morgan Algebras. In: Ehrenfeucht, A., Grzegorczyk, A., Mycielski, J., Ryll-Nardzewski, C., Eds., Tren ds in Logic III , Studia Logica, Warsaw, 23-25 September 2005. https://mimuw.edu.pl/~mrr/
Belyavskaya, G.B. and Cheban, A.M. (1972) S-Systems of Arbitrary Index I. Mat Issledovaniya Akad . Nauk Moldavii , Kishinev , 7, 23-43.
Belyavskaya, G.B. and Cheban, A.M. (1972) S-Systems of Arbitrary Index II. Mat Issledovaniya Akad . Nauk Moldavii , Kishinev , 7, 3-13.
Bloom, S.L., Ésik, Z. and Manes, E.G. (1990) A Cayley Theorem for Boolean Algebras. The American Mathematical Monthly , 97, 831-833. https://doi.org/10.1080/00029890.1990.11995668
Movsisyan, Y.M. (2009) Binary Representations of Algebras with at Most Two Binary Operations: A Cayley Theorem for Distributive Lattices. International Journal of Algebra and Computation , 19, 97-106. https://doi.org/10.1142/s0218196709004993
Padmanabhan, R. and Penner, P. (1998) Lattice Ordered Polynomial Algebras. Order , 15, 75-86. https://doi.org/10.1023/a:1006036222760
Przytycki, J.H. (2015) Knots and Distributive Homology: From Arc Colorings to Yang-Baxter Homology. In: Kauffman, L.H. and Manturov, V.O., Eds., New Ideas in Low Dimensional Topology , World Scientific, 413-488. https://doi.org/10.1142/9789814630627_0011
Stein, S.K. (1957) On the Foundations of Quasigroups. Transactions of the American Mathematical Society , 85, 228-256. https://doi.org/10.1090/s0002-9947-1957-0094404-6
Urbanik, K. (1964) On Algebraic Operations in Idempotent Algebras. Colloquium Mathematicum , 13, 129-157. https://doi.org/10.4064/cm-13-2-129-157
Lau, D. (2006) Functional Algebras on Finite Sets, a Basic Course on Many-Valued Logic and Clone Theory. Springer-Verlag.
Post, E.L. (1921) Introduction to a General Theory of Elementary Propositions. American Journal of Mathematics , 43, 163-185. https://doi.org/10.2307/2370324
Post, E. (1941) The Two-Valued Iterative Systems of Mathematical Logic. Princeton University Press.
Arieli, O. and Avron, A. (1998) The Value of the Four Values. Artificial Intelligence , 102, 97-141. https://doi.org/10.1016/s0004-3702(98)00032-0
Belnap, N.D. (1977) A Useful Four-Valued Logic. In: Epstein, G. and Dunn, J.M., Eds., Modern Uses of Multiple-Valued Logic , Springer Netherlands, 5-37. https://doi.org/10.1007/978-94-010-1161-7_2
Bou, F. and Rivieccio, U. (2010) The Logic of Distributive Bilattices. Logic Journal of IGPL , 19, 183-216. https://doi.org/10.1093/jigpal/jzq041
Gehrke, M., Walker, C. and Walker, E. (1997) A Mathematical Setting for Fuzzy Logics. International Journal of Uncertainty , Fuzziness and Knowledge-Based Systems , 5, 223-238. https://doi.org/10.1142/s021848859700021x
Gehrke, M., Walker, C. and Walker, E. (2000) Some Comments on Fuzzy Normal Forms. Ninth IEEE International Conference on Fuzzy Systems . FUZZ — IEEE 2000 ( Cat . No .00 CH 37063), San Antonio, 7-10 May 2000, 593-598. https://doi.org/10.1109/fuzzy.2000.839060
Kondo, M. (2002) On the Structure of Weak Interlaced Bilattice. Proceedings 32 nd IEEE International Symposium on Multiple - Valued Logic , Boston, 15-18 May 2002, 23-26. https://doi.org/10.1109/ismvl.2002.1011065
Mobasher, B., Pigozzi, D. and Slutzki, G. (1997) Multi-Valued Logic Programming Semantics an Algebraic Approach. Theoretical Computer Science , 171, 77-109. https://doi.org/10.1016/s0304-3975(96)00126-0
Mobasher, B., Pigozzi, D., Slutzki, G. and Voutsadakis, G. (2000) A Duality Theory for Bilattices. Algebra Universalis , 43, 109-125. https://doi.org/10.1007/s000120050149