On the Pecking Order Between Those of Mitsch and Clifford
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Vikash Kumar Gupta
ABSTRACT
Order-theoretic explorations of algebraic structures are known to lead to hitherto hidden insights. Two such relations that have stood out are those of Mitsch and Clifford – the former for the generality in its application and the latter for the insights it offers. In this work, our motivation is to study the converse: we want to explore the extent of the utility of Mitsch’s order and the applicability of Clifford’s order. Firstly, we show that if the Mitsch’s poset is either bounded or a chain, arguably a richer order theoretic structure, the semigroup reduces to one of a simple band. Secondly, noting that the special semigroups on which Clifford’s relation does give rise to an order has not been characterised so far, we solve this problem by proposing a property called Quasi-Projectivity that is essential in this context and also give necessary and sufficient conditions for the Clifford’s relation to give a total and compatible order, even if the semigroup is not commutative. Further, by showing some interesting connections between this relation and the orders obtained by Green’s relations, we further reaffirm the importance and naturalness of the order proposed by Clifford. Finally, by discussing the Clifford’s relations on ordered semigroups, we present some novel perspectives and also show that some of the assumptions in the often cited results of Clifford’s are not necessary. On the whole, our study argues favourably towards Clifford’s than that of the Mitsch’s relation, in so far as the structural information gained about the underlying semigroup.
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Also see Section 6.
- 2
For an interesting review-cum-commentary of his works, we refer the readers to [12].
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Note that by a Mitsch poset we refer to any semigroup with the order on it defined by
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A semigroup that has an identity element e, has no non-trivial invertible elements and satisfies the left cancellation law [17].
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See Drazin [9] for the definitions and relations among these special semigroups.
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A part of this result has been proven in [11] in the context where S = [0, 1].
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To remain consistent with the notations we should have used
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- 9
In Section 7.1 we show that even when the Clifford’s order is not total the results may still hold.
- 10
Please see Section 7.2 for the definitions of these terms.
- 11
A monoid where the operation is compatible w.r.t. the order and the identity is also the top element.
REFERENCES
[1] Blyth, T. S.—Gomes, G. M. S.: On the compatibility of the natural order on a regular semigroup, Proc. Roy. Soc. Edinburgh Sect. A: Mathematics 94 (1983), 79–84.10.1017/S0308210500016140Search in Google Scholar
[2] Clifford, A. H.: Arithmetic and ideal theory of commutative semigroups, Ann. of Math. 39 (1938), 594–610.10.2307/1968637Search in Google Scholar
[3] Clifford, A. H.: Partially ordered Abelian groups, Ann. of Math. 41 (1940), 465–473.10.2307/1968728Search in Google Scholar
[4] Clifford, A. H.: Naturally totally ordered commutative semigroups, Amer. J. Math. 76 (1954), 631–646.10.2307/2372706Search in Google Scholar
[5] Clifford, A. H.: Connected ordered topological semigroups with idempotent endpoints, Trans. Amer. Math. Soc. 88 (1958), 80–98.10.1090/S0002-9947-1958-0095221-4Search in Google Scholar
[6] Clifford, A. H.: Ordered commutative semigroups of the second kind, Proc. Amer. Math. Soc. 9 (1958), 682–687.10.1090/S0002-9939-1958-0096740-2Search in Google Scholar
[7] Clifford, A. H.: Totally ordered commutative semigroups, Bull. Amer. Math. Soc. 64 (1958), 305–316.10.1090/S0002-9904-1958-10221-9Search in Google Scholar
[8] Conrad, P. F.: The hulls of semiprime rings, Bull. Aust. Math. Soc. 12 (1975), 311–314.10.1017/S0004972700023911Search in Google Scholar
[9] Drazin, M. P.: A partial order in completely regular semigroups, J. Algebra 98 (1986), 362–374.10.1016/0021-8693(86)90003-7Search in Google Scholar
[10] Green, J. A.: On the structure of semigroups, Ann. of Math. 54 (1986), 163–172.10.2307/1969317Search in Google Scholar
[11] Gupta, V. K.—Jayaram, B.: Order based on associative operations, Information Sciences 566 (2021), 326–346.10.1016/j.ins.2021.02.020Search in Google Scholar
[12] Hofmann, K. H.—Lawson, J. D.: Linearly ordered semigroups: Historical origins and A. H. Clifford’s influence, Semigroup Theory and its Applications (K. H. Hofmann and W. Mislove, eds.), Cambridge University Press, Cambridge, 1986, pp. 163-172.Search in Google Scholar
[13] Klement, E. P.—Mesiar, R.—Pap, E.: Triangular norms as ordinal sums of semigroups in the sense of A. H. Clifford, Semigroup Forum 65 (2002), 71–82.10.1007/s002330010127Search in Google Scholar
[14] Mitsch, H.: A natural partial order for semigroups, Proc. Amer. Math. Soc. 97 (1986), 384–388.10.1090/S0002-9939-1986-0840614-0Search in Google Scholar
[15] Mitsch, H.: Semigroups and their natural order, Math. Slovaca 44 (1994), 445–462.Search in Google Scholar
[16] Nambooripad, K. S.: The natural partial order on a regular semigroup, Proc. Edinb. Math. Soc. 23 (1980), 249–260.10.1017/S0013091500003801Search in Google Scholar
[17] Siegrist, K.: Probability Distributions on Partially Ordered Sets and Positive Semigroups, Lecture Notes, 1980.Search in Google Scholar
[18] Steinberg, B.: Inverse semigroup homomorphisms via partial group actions, Bull. Aust. Math. Soc. 64 (2001), 157–168.10.1017/S0004972700019778Search in Google Scholar
[19] Sussman, I.: A generalization of Boolean rings, Math. Ann. 136 (1958), 326–338.10.1007/BF01360238Search in Google Scholar
[20] Vagner, V. V.: The theory of generalized heaps and generalized groups, Matematicheskii Sbornik 74 (1953), 545–632.Search in Google Scholar
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Articles in the same Issue
- Parameters in Inversion Sequences
- On the Pecking Order Between Those of Mitsch and Clifford
- A Note on Cuts of Lattice-Valued Functions and Concept Lattices
- Trigonometric Sums and Riemann Zeta Function
- Notes on the Equation d(n) = d(φ(n)) and Related Inequalities
- Harmony of Asymmetric Variants of the Filbert and Lilbert Matrices in q-form
- Maximal Density and the Kappa Values for the Families {a, a + 1, 2a + 1, n} and {a, a + 1, 2a + 1, 3a + 1, n}
- Extensions in Time Scales Integral Inequalities of Jensen’s Type via Fink’s Identity
- A Note on Fractional Simpson Type Inequalities for Twice Differentiable Functions
- Extended Bromwich-Hansen Series
- Iterative Criteria for Oscillation of Third-Order Delay Differential Equations with p-Laplacian Operator
- Vallée-Poussin Theorem for Equations with Caputo Fractional Derivative
- On Oscillatory Behavior of Third Order Half-Linear Delay Differential Equations
- On Existence and Uniqueness of Solutions for Ordinary Differential Equations in Locally Convex Topological Linear Spaces
- Bounds on Blow-Up Time for a Higher-Order Non-Newtonian Filtration Equation
- On a Solvable System of Difference Equations in Terms of Generalized Fibonacci Numbers
- The Smallest and the Largest Families of Some Classes of 𝒜-Continuous Functions
- Type II Exponentiated Half-Logistic Gompertz-G Family of Distributions: Properties and Applications
- Strong Consistency of Least-Squares Estimators in the Simple Linear Errors-in-Variables Regression Model with Widely Orthant Dependent Random Variables
Articles in the same Issue
- Parameters in Inversion Sequences
- On the Pecking Order Between Those of Mitsch and Clifford
- A Note on Cuts of Lattice-Valued Functions and Concept Lattices
- Trigonometric Sums and Riemann Zeta Function
- Notes on the Equation d(n) = d(φ(n)) and Related Inequalities
- Harmony of Asymmetric Variants of the Filbert and Lilbert Matrices in q-form
- Maximal Density and the Kappa Values for the Families {a, a + 1, 2a + 1, n} and {a, a + 1, 2a + 1, 3a + 1, n}
- Extensions in Time Scales Integral Inequalities of Jensen’s Type via Fink’s Identity
- A Note on Fractional Simpson Type Inequalities for Twice Differentiable Functions
- Extended Bromwich-Hansen Series
- Iterative Criteria for Oscillation of Third-Order Delay Differential Equations with p-Laplacian Operator
- Vallée-Poussin Theorem for Equations with Caputo Fractional Derivative
- On Oscillatory Behavior of Third Order Half-Linear Delay Differential Equations
- On Existence and Uniqueness of Solutions for Ordinary Differential Equations in Locally Convex Topological Linear Spaces
- Bounds on Blow-Up Time for a Higher-Order Non-Newtonian Filtration Equation
- On a Solvable System of Difference Equations in Terms of Generalized Fibonacci Numbers
- The Smallest and the Largest Families of Some Classes of 𝒜-Continuous Functions
- Type II Exponentiated Half-Logistic Gompertz-G Family of Distributions: Properties and Applications
- Strong Consistency of Least-Squares Estimators in the Simple Linear Errors-in-Variables Regression Model with Widely Orthant Dependent Random Variables
