Skip to main content
Log in

l-Hemi-Implicative Semilattices

  • Published:
Studia Logica Aims and scope Submit manuscript


An l-hemi-implicative semilattice is an algebra \(\mathbf {A} = (A,\wedge ,\rightarrow ,1)\) such that \((A,\wedge ,1)\) is a semilattice with a greatest element 1 and satisfies: (1) for every \(a,b,c\in A\), \(a\le b\rightarrow c\) implies \(a\wedge b \le c\) and (2) \(a\rightarrow a = 1\). An l-hemi-implicative semilattice is commutative if if it satisfies that \(a\rightarrow b = b\rightarrow a\) for every \(a,b\in A\). It is shown that the class of l-hemi-implicative semilattices is a variety. These algebras provide a general framework for the study of different algebras of interest in algebraic logic. In any l-hemi-implicative semilattice it is possible to define an derived operation by \(a \sim b := (a \rightarrow b) \wedge (b \rightarrow a)\). Endowing \((A,\wedge ,1)\) with the binary operation \(\sim \) the algebra \((A,\wedge ,\sim ,1)\) results an l-hemi-implicative semilattice, which also satisfies the identity \(a \sim b = b \sim a\). In this article, we characterize the (derived) commutative l-hemi-implicative semilattices. We also provide many new examples of l-hemi-implicative semilattice on any semillatice with greatest element (possibly with bottom). Finally, we characterize congruences on the classes of l-hemi-implicative semilattices introduced earlier and we characterize the principal congruences of l-hemi-implicative semilattices.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Ardeshir, M., and W. Ruitenburg, Basic propositional calculus I, Mathematical Logic Quarterly 44: 317–343, 1998.

    Article  Google Scholar 

  2. Busneag, D., and M. Ghita, Some latticial properties of Hilbert algebras, Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie 53: 87–107, 2010.

    Google Scholar 

  3. Celani, S. A., and R. Jansana, Bounded distributive lattices with strict implication, Mathematical Logic Quarterly 51(3): 219–246, 2005.

    Article  Google Scholar 

  4. Cīrulis, J., Weak relative pseudocomplements in semilattices, Demostratio Mathematica XLIV(4): 651–672, 2011.

    Google Scholar 

  5. Ciungu, L. C., On pseudo equality algebras, Archive for Mathematical Logic 53(5): 561–570, 2014.

    Article  Google Scholar 

  6. Cornejo, J. M., On some semi-intuitionistic logics, Studia Logica 103: 303–344, 2015.

    Article  Google Scholar 

  7. Curry, H. B., Foundations of Mathematical Logic. McGraw-Hill, New York, 1963.

    Google Scholar 

  8. Dvurecenskij, A., and O. Zahir, Pseudo equality algebras: revision, Soft Computing 20(6): 2091–2101, 2016.

    Article  Google Scholar 

  9. Figallo, A. V., G. Z. Ramón, and S. Saad, A note on the Hilbert algebras with infimum, Matematica Contemporary 24: 23–37, 28th Workshop on Logic, Language, Informations and Computation, WoLLIC’2001, Brasilia, 2003.

  10. Font, J. M., On semilattice-based logics with an algebraizable assertional companion, Reports on Mathematical Logic 46: 109–132, 2011.

    Google Scholar 

  11. Idziak, P. M., Lattice operations in BCK-algebras, Mathematica Japonica 20: 839–846, 1984.

    Google Scholar 

  12. Jenei, S., Equality algebras, Studia Logica 100: 1201–1209, 2012.

    Article  Google Scholar 

  13. Jenei, S., and L. Kóródi, Pseudo equality algebras, Archive for Mathematical Logic 52(3): 469–481, 2013.

    Article  Google Scholar 

  14. Meng, J., Y. B. Jun, and S. M. Hong, Implicative semilattices are equivalent to positive implicative BCK-algebras with condition (S), Mathematica Japonicae 48: 251–255, 1998.

    Google Scholar 

  15. Nemitz, W., Implicative semi-lattices, Transactions of the American Mathematical Society 117: 128–142, 1965.

    Article  Google Scholar 

  16. Sankappanavar, H. P., Semi-Heyting algebras: an abstraction from Heyting algebras, in Proceedings of the 9th Congreso “Dr. Antonio A. R.”, 33–66, Actas Congr. “Dr. Antonio A. R. Monteiro”, Universidad Nacional del Sur, Bahía Blanca, Argentina, 2008.

  17. Sankappanavar, H. P., Expansions of semi-heyting algebras I: discriminator varieties, Studia Logica 98 (1–2): 27–81, 2011.

    Article  Google Scholar 

  18. San Martín, H. J., Compatible operations on commutative weak residuated lattices, Algebra Universalis 73(2): 143–155, 2015.

    Article  Google Scholar 

  19. San Martín, H. J., On congruences in weak implicative semi-lattices, Soft Computing 21(12): 3167–3176, 2017.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Hernán Javier San Martín.

Additional information

Presented by Jacek Malinowski

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Castiglioni, J.L., San Martín, H.J. l-Hemi-Implicative Semilattices. Stud Logica 106, 675–690 (2018).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: