Narrow Results

Anti-elementarity is a strong way of ensuring that a class of structures, in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form ${ℒ}_{\infty \lambda }$. We prove that many naturally defined classes are anti-elementary, including the following:

• the class of all lattices of finitely generated convex $\ell$-subgroups of members of any class of $\ell$-groups containing all Archimedean $\ell$-groups;
• the class of all semilattices of finitely generated $\ell$-ideals of members of any nontrivial quasivariety of $\ell$-groups;
• the class of all Stone duals of spectra of MV-algebras — this yields a negative solution to the MV-spectrum Problem;
• the class of all semilattices of finitely generated two-sided ideals of rings;
• the class of all semilattices of finitely generated submodules of modules;
• the class of all monoids encoding the nonstable K₀-theory of von Neumann regular rings, respectively, C*-algebras of real rank zero;
• (assuming arbitrarily large Erdős cardinals) the class of all coordinatizable sectionally complemented modular lattices with a large $4$-frame.

The main underlying principle is that under quite general conditions, for a functor $\mathrm{\Phi }:{𝒜}\to {ℬ}$, if there exists a noncommutative diagram $\overrightarrow{D}$ of ${𝒜}$, indexed by a common sort of poset called an almost join-semilattice, such that

• $\mathrm{\Phi }{\overrightarrow{D}}^I$ is a commutative diagram for every set $I$,
• $\mathrm{\Phi }\overrightarrow{D}\ncong \mathrm{\Phi }\overrightarrow{X}$ for any commutative diagram $\overrightarrow{X}$ in ${𝒜}$,

then the range of $\mathrm{\Phi }$ is anti-elementary.

A novel colorimetric indicator for pork freshness composed of agar, ZnTPPS₄ and glycerin was designed and developed by casting/solvent evaporation method. The FT-IR, UV-vis, XRD and SEM were employed to analyze the structure and valence bonds of ZnTPPS₄/AG compound, and the results showed good compatibility between agar and ZnTPPS₄. The tensile strength and elongation at the breaking point of indicator films increased slightly with increase of porphyrin, which may have been a contributing factor of H-bonds. After 7 days of placement, the total color variation ($\mathrm{\Delta }$E) of major films was less than 5, manifesting in the color stability being sufficient enough to act as a color indicator, and the mechanism for color variation was explained therein. Furthermore, the prepared films were utilized as indicators for monitoring the freshness of lean meat at room temperature (25${}^{\circ }$C). The total volatile basic nitrogen (TVB-N) of pork and $\mathrm{\Delta }$E of the labels were recorded simultaneously. The trials demonstrated that ZnTPPS₄/AG with higher content of ZnTPPS₄ had the superior sensitivity and the color changes of labels in pork packaging changed according to the decay threshold of TVB-N, which implied that ZnTPPS₄/AG was able to indicate the spoilage via colorimetric method. Therefore, these novel indication labels could be used to monitor the pork freshness in a real-time, nondestructive and inexpensive way.

Graph theory is a useful tool to study the properties of algebraic structures. Here we are going to employ this tool to study nominal sets. To do so, we employ the freshness relation to construct certain graphs so-called fresh-graph, due to the key role of freshness relation in obtaining these graphs. We then investigate the properties of these graphs to determine some properties of nominal sets.