Subject / 7 records
Lattices and ordered algebra
Free lattices, Heyting spectra, residuation, amalgamation, and ordered algebraic structures.
Classification: Second Brain
Explore this area on the research map →Connected through shared records
Reading routes
7 records
- 10.5281/zenodo.234770162026
Ascending chains of sublattices in free lattices
Martino Lupini · Mathematics
We prove that every increasing sequence of free sublattices of bounded finite rank in a free lattice is eventually constant. In particular, there is no infinite strictly increasing chain of copies of the free lattice on three generators, answering negatively a question of R. A. Dean recorded in 1997. Freeness is essential: we construct in the three-generated free lattice an infinite strictly increasing chain of pairwise nonisomorphic retracts, each requiring exactly three generators. The corresponding retractions commute. We also give a proper self-embedding whose fixed-point sublattice contains a free copy of the ambient lattice, and prove that the stable image of every endomorphism of a finitely generated free lattice consists precisely of its periodic points.
PDF · LaTeX · GitHub named
- 10.5281/zenodo.234696252026
Forests as Heyting spectra: representation criteria, constructions, and obstructions
Martino Lupini · Orderability
Our main result shows that an Esakia space whose order is a downward forest has a locally finite clopen-upset algebra whenever its maximal boundary is scattered. More precisely, failure of local finiteness forces a nonempty perfect principal maximal boundary. We also construct locally finite representations of immediately splitting trees without assuming well-foundedness, and classify the cardinal multiplicities of a uniform family of binary canopies. The countable-forest inverse-sequence characterization and failure of infinitary order axiomatizability are consequences of these results, using established inverse-limit and model-theoretic methods. We give finite-partition certificates as explicit forms of the existing duality and compactification machinery. The unrestricted intrinsic structural classification of forest spectra remains unresolved here.
PDF · LaTeX · GitHub named
- 10.5281/zenodo.234196352026
Saturation and equational hulls of ordered semigroup varieties
Martino Lupini · Orderability
We prove that an inequality-defined variety of partially ordered semigroups is saturated if and only if the ordinary semigroup variety generated by its reducts is saturated. Here saturation excludes proper epimorphic extensions in the ambient category of all semigroups or all partially ordered semigroups. This transfers ordinary equational saturation criteria to compatible partial orders without finiteness, permutation, or convexity assumptions. It recovers a positive answer to Ahanger, Shah, and Khan’s question about ordered normal bands and gives a general equational framework for relaxing convexity in related saturation results. The proof uses the completely regular core E(S)SE(S), a bounded-exponent variety symmetry theorem, and a countable construction for arbitrary inequational theories.
PDF · LaTeX · GitHub named
- 10.5281/zenodo.234195772026
Amalgamation in an infinite chain of commutative idempotent residuated-lattice varieties
Martino Lupini · Mathematics
We construct a strictly increasing infinite chain of finitely generated varieties of commutative idempotent residuated lattices, each with the amalgamation property. The generator of Vₙ is a quasiprimal (n+3)-element algebra obtained by adjoining an absorbing least element to a bounded lattice with n incomparable interior elements; multiplication is join elsewhere. Homogeneity, congruence extension, and the classical homomorphism-extension method yield amalgamation. For n ≥ 2 the varieties are not semilinear, and for n ≥ 3 they are not lattice-distributive. This answers negatively the unrestricted finiteness question posed by Fussner, Metcalfe, and Santschi in Interpolation and the Exchange Rule, circulated in 2023 and published in 2026. The distributive version remains open. Among finite Stanovský algebras L⁰, amalgamation holds exactly when the interior of L is an antichain. Strong amalgamation holds for V₀ and fails for every Vₙ with n ≥ 1.
PDF · LaTeX · Python · Data · GitHub named
- 10.5281/zenodo.234193932026
The variety of integral residuated lattices fails the amalgamation property
Martino Lupini · Mathematics
We answer negatively the open question whether integral residuated lattices have the amalgamation property, recorded by Metcalfe, Paoli, and Tsinakis in their 2023 monograph and explicitly posed by Jipsen and Santschi in Residuated lattices do not have the amalgamation property. A span of finite integral residuated lattices, with a three-element common algebra and extensions of sizes five and six, has no amalgam even among all residuated lattices. Thus the three-element Gödel chain, in the signature without a bottom constant, is not an amalgamation base. The same example excludes amalgamation in every variety containing the five-element chain, including the varieties of n-potent integral residuated lattices for all n ≥ 2. We also isolate a local obstruction without global integrality assumptions and characterize amalgamation of a common-bottom quotient-product span by extension of the corresponding congruence.
PDF · LaTeX · Python · Data · GitHub named
- 10.5281/zenodo.232775982026
Distributivity of lattice-ordered pregroups
Martino Lupini · Orderability
No abstract on this deposit.
PDF · LaTeX
- 10.5281/zenodo.231050332026
An eight-element parachute lattice and its dual: finite representations with two unary operations
Martino Lupini · Mathematics
No abstract on this deposit.
PDF · LaTeX · Notes · Data