Simplified axiomatic system of DRl-semigroups
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Tomáš Kovář
Abstract
DRl-semigroups are a generalization of lattice ordered groups (l-groups) containing a.o. Boolean algebras, Brouwerian algebras and MV-algebras. Since their introduction in the 1960s their axiomatic system has been several times reduced.
In this paper, we show that it can be simplified even further. Specifically, we show that the axiom ensuring compatibility of the semigroup operation + and the lattice operations is equivalent to a significantly weaker condition of monotonicity of + with respect to the implied order ≤. Because the same equivalence holds for l-groups, we observe that the axiomatic systems of l-groups and DRl-semigroups are more aligned than originally thought.
(Communicated by Anatolij Dvurečenskij)
References
[1] Kovář, T.: Note on axioms of a dually residuated lattice ordered semigroup, Discuss. Math. Algebra Stochastic Methods 17 (1997), 89–90.Search in Google Scholar
[2] Kovář, T.: Two remarks on dually residuated lattice ordered semigroups, Math. Slovaca 49 (1999), 17–18.Search in Google Scholar
[3] Swamy, K. L. N.: Dually residuated lattice ordered semigroups, Math. Ann. 159 (1965), 105–114.10.1007/BF01360284Search in Google Scholar
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Articles in the same Issue
- A note on the coprime power graph of groups
- Simplified axiomatic system of DRl-semigroups
- Special filters in bounded lattices
- Reichenbach’s causal completeness of quantum probability spaces
- A construction of magmas and related representation
- Extensions of the triangular D(3)-Pair {3, 6}
- Hermite-Hadamard type inequalities for new class h-convex mappings utilizing weighted generalized fractional integrals
- Divergence operator of regular mappings
- Monotonicity of the ratio of two arbitrary gaussian hypergeometric functions
- Oscillation and asymptotic criteria for certain third-order neutral differential equations involving distributed deviating arguments
- Multiplicity results for a fourth-order elliptic equation of p(x)-kirchhoff type with weights
- Singular discrete dirac equations
- Convergence of bivariate exponential sampling series in logarithmic weighted spaces of functions
- Fundamental inequalities for the iterated Fourier-cosine convolution with Gaussian weight and its application
- Existence of solutions and Hyers-Ulam stability for κ-fractional iterative differential equations
- On almost cosymplectic generalized (k, μ)ʹ-spaces
- On some recent selective properties involving networks
- Minimal usco and minimal cusco maps and the topology of pointwise convergence
- Corrigendum to: Every positive integer is a sum of at most n + 2 centered n-gonal numbers
